V. 


\  ■-■^\\ 


0 


trCSB  IIBRART 


t. 


1.    - 


■~±X-  (^-i^^-h. 


a:  '^  = 


it 


« 


AN 


ELEiMENTARY   TREATISE 


M  E  C  HAN  I  C  S, 


COMPREHENDING 


THE   DOCTRINE   OF   EQUILIBRIUM  AND  MOTION, 


AS    APPLIED    TO 


SOLIDS    AND    FLUIDS, 


CHIEFLY    COMPILED,    AND    DESIGNED 


FOR  THE  USE  OF  THE  STUDENTS  OF  THE  UNIVERSITY 


CAMBRIDGE,  NEW  ENGLAND. 


By   JOHN    FARRAR,   LL.  D., 

HOLLIS    PROFESSOR    OF    MATHEMATICS    AND    NATURAL    PHILOSOPHY. 


SECOND    EDITION,    REVISED    AND    CORRECTED. 


BOSTON: 
IJ  I  L  L  I  A  R  D  ,  GRAY,  AND  COMPANY 

1  >J  :i  4 . 


Entered  according  to  act  of  Congress,  in  the  year  1834, 
by  John  Farrar, 
in  the  Clerk's  office  of  the  District  Court  of  the  District  of  Massachusetts. 


CAMBRIDGE: 
CHARLES    FOLSOM,    PRINTER   TO    THE    UNIVERSITY. 


ADVERTISEMENT. 


In  selecting  materials  for  this  treatise  particular  re- 
gard has  been  had  to  the  practical  uses  of  the  science ; 
at  the  same  time  the  theoretical  principles  are  rigorously 
demonstrated.  Where  the  nature  of  the  subject  ad- 
mitted of  it,  the  geometrical  method  has  been  pre- 
ferred, as  being  more  perspicuous,  and  better  adapted  to 
most  learners.  There  are  many  refinements  in  the 
later  and  more  improved  treatises  not  adopted  in  this, 
for  the  reasons  above  mentioned,  and  on  account  of  the 
insufficient  provision,  as  to  time  and  preparatory  studies, 
that  is  made  in  most  of  the  seminaries  of  the  United 
States  for  a  text-book  upon  such  a  plan. 

The  works  principally  used  in  preparing  this  treatise 
are  those  of  Biot,  Bezout,  Poisson,  Francoeur,  Gregory, 
Whewell,  and  Leshe.  In  the  portions  selected  it  was 
found  necessary  to  make  considerable  alterations  and 
additions  in  order  to  give  a  uniform  character  to  the 
w^hole.  There  has  often  been  occasion,  moreover,  in 
appropriating  the  substance  of  a  proposition  or  course  of 
reasoning,  to  amplify  or  condense  it,  or  to  vary  the 
phraseology.  It  became  inconvenient,  therefore,  to  dis- 
tinguish by  quotations  the  respective  portions  taken 
from  different  authors.  Bezout  has  been  adopted,  in 
substance,  as  the  basis  in  what  relates  to  statics,  dynam- 
ics, and  hydrostatics,   although  the  matter  is  arranged 


IV  ADVERTISEMENT. 

according  to  a  different  system  ;  and  Gregory,  with  many 
changes  and  substitutions,  has  been  principally  used  in 
that  which  is  comprehended  under  hydrodynamics. 

This  volume  constitutes  the  first  part  of  a  course  of 
Natural  Philosophy.  The  second  comprehends  Elec- 
tricity, Magnetism,  and  Electro-magnetism ;  the  third 
Optics,  and  the  fourth  Astronomy.  These  subjects  are 
treated  in  three  different  volumes. 

Cambridge,  Massachusetts, 
September  30th,  1834. 


CONTENTS. 


Preliminary  Remarks  and  Definitions 


Page. 


STATICS. 

Uniform  Motion            .             .             .             .             .  ,13 

Forces  and  the  Quantity  of  Motion            ...  16 
Equilibrium  between  Forces  directly  opposite                .  .18 
Compound  Motion                .             •             •             .             .  20 
Composition  and  Decomposition  of  Forces       .              ,  .24 
Moments  and  their  Use  in  the  Composition  and  Decomposi- 
tion of  Forces.             .             .              .             .             .  31 

Parallel  Forces  ivhich  act  in  different  Planes                .  .       38 
Forces,    the    Directions   of  which   are  neither  in  the  same 

Plane  nor  parallel  to  each  other           .             .             .  41 

Centre  of  Gravity        .             .             .             .             .  .43 

Application  of  the   Principles  of  the  Centre  of  Gravity  to 

particular  Problems     .....  50 

Properties  of  the  Centre  of  Gravity                .              .  .69 
General  Principle  of  the  Equilibrium  of  Bodies                .  75 
D' Alembcrt' s  Principle,  and  Concluding'  Deductions  .       76 
Application  of  the   Principles  of  Equilibrium  to   the   Ma- 
chines usually  denominated  Mechanical  Powers            .  79 
Rope  Machine               .             .             .             .             .  .79 

Lever                        ......  90 

Pulley                .             .              .              .              .              .  .102 

Wlieel  and  Axle      .             .                          .             .             .  108 

Inclined  Plane              .             •             .             .             .  .116 

Screw          .......  124 

Wedge               ...              •             .              .  .     128 

General  Law  of  Equilibrium  in  Machines                           .  130 


VI 


CONTENTS. 


Miction 

Stiffness  of  Cords 
Balance,  Steelyard,  4*c. 


135 
155 

160 


DYNAMICS. 

Motion  uniformly  accelerated        ....  165 

Free  Motion  in  heavy  Bodies              .             .             .  .168 

Direct  Collision  of  Bodies            ....  175 

Direct  Collision  of  unelastic  Bodies               .             .  .       176 

Force  of  Inertia                 .             •             •             •             .  178 
Application  of  the  Principles  of  Collision  of  unelastic  Bodies  181 

Collision  of  elastic  Bodies             .             .              .              .  185 

Motion  of  Projectiles              .              .             .             .  .190 

Motion  of  heavy  Bodies  down  inclined  Planes     .             .  205 

Motion  along  curved  Surfaces             .             .             .  .210 

Motion  of  Oscillation       .....  215 

Pendulums     .             .             .             .             .             •  .221 

Line  of  swiftest  Descent               ....  225 

Moment  of  Inertia     ......       230 

Centre  of  Percussion  and  Centre  of  Oscillation                .  236 

Actual  Length  of  the  Seconds  Pendulum      .              .  .       246 

Measure  of  the  Force  of  Gravity             .             .             .  249 

Application  of  the  Pendulum  to  Time-Keepers        .  .       250 

Rotation  of  Bodies  unconfned     ....  257 

Method  of  estimating  the  Forces  applied  to  Machines  .       265 

Maximum  Efect  of  Agents  and  Machines            .             .  273 


HYDROSTATICS. 

Introductory  Remarks.             ....  289 

Pressure  of  Fluids    ......  290 

Solids  immersed  in  Fluids             ....  306 

Specific  Gravities       .             .             .             .              .              .  308 

Spirit  Level           .             .             .             .             .  321 

Equilibrium  of  Floating  Bodies        ....  322 

Capillary  Attraction         .....  336 

Apparent  Attraction  and  Repulsion  observed  in  Bodies  float- 
ing near  each  other  on  the  Surface  of  Fluids     .              .  349 
Barometer              ......  351 


CONTENTS. 


HYDRODYNAMICS. 


Discharge  of  Fluids  through  Apertures  in  the  Bottom  and 

Sides  of  Vessels  .  .  .  .  .369 

Motion  of  Gases                .....  377 

Resistance  of  Fluids  to  Bodies  moving  in  them         .             .  388 

Theory  of  the  Air-Pump              ....  399 

Pumps  for  Raising  Water    .             .             .             <            .  402 

Syphon     .......  412 

Steam-Engine            ......  413 

NOTES              ......  427 


AN 


ELEMENTARY  TREATISE 


MECHANICS. 


Preliminary  Remarks  and  Definitions. 

1.  Matter  has  been  variously  defined  by  philosophers,  and 
some  have  even  doubted  whether  we  can  be  morally  certain  of  its 
existence.  It  is  not  our  intention,  nor  does  it  belong  to  the  nature 
of  our  subject,  to  enter  into  discussions  of  this  kind.  Relying 
solely  upon  experiment,  we  give  the  name  of  matter  or  body  to 
whatever  is  capable  of  producing,  through  our  organs,  certain 
determinate  sensations ;  and  the  power  of  exciting  in  us  these  sen- 
sations constitutes  for  us  so  many  properties,  by  which  we  re- 
cognise the  presence  of  bodies.  But  among  these  properties, 
two  only  are  absolutely  essential  in  order  to  our  having  a  percep- 
tion of  matter.  These  are  extension  and  impenetrahiliiy ,  of  which 
the  sight  and  touch  are  the  first  judges. 

2.  The  character  derived  from  extension  is  self-evident  j  when 
we  see  or  touch  a  body,  this  body,  or,  if  you  please,  the  power 
which  it  has  of  affecting  us,  resides  in  a  certain  portion  of  space. 
The  place  which  it  occupies  is  therefore  determinate  ',  and  by  this 
very  circumstance  it  is  extended. 

3.  When  we  pass  our  hands  over  the  surface  of  a  body,  we  per- 
ceive that  the  matter  of  which  it  is  composed,  is  without  us ;  more- 
over, two  distinct  portions  of  matter  can  never  be  made  to  co- 
incide or  identify  themselves,  the  one  with  the  other,  in  such  a 
manner  that  the  same  absolute  points  of  space  shall  at  the  same 
time  give  us  the  sensation  of  both.  In  this  consists  the  property 
of  impenetrability. 

1 


2  Preliminary  Remarks. 

To  show  how  this  property,  together  with  that  of  extension,  is 
necessary  to  constitute  a  hotly,  I  will  refer  to  familiar  phenomena  in 
which  these  properties  are  observed  separately. 

If  an  object  he  placed  before  a  concave  mirror,  there  will  be 
formed,  at  a  certain  distance  from  the  mirror,  an  image  of  the  object. 
This  image,  distinct  from  the  parts  of  space  that  surround  it,  is 
extended  but  not  impenetrable.  The  hand  may  be  thrust  through 
it  without  experiencing  the  smallest  resistance,  and  the  parts  that 
come  in  contact  with  the  hand,  vanish  instead  of  being  displaced. 
A  piece  of  wood  or  stone  does  not  admit  of  being  thus  penetrated. 
Moreover,  by  means  of  a  second  mirror  properly  disposed,  an 
image  of  another  object  may  be  made  to  occupy  the  same  place 
with  that  of  the  first,  without  the  latter  being  displaced,  or  in  any 
way  deranged.  Indeed  the  same  coincidence  may  be  effected  with 
a  third,  a  fourth,  or  any  number  of  images.  These  images  are  ex- 
tended, but  not  impenetrable  ;  they  are  forms,  but  not  sensible  mat- 
ter. I  say  sensible  matter,  for  we  shall  see  hereafter  that  light 
which  constitutes  these  images,  is  itself  probably  composed  of  ma- 
terial particles  of  an  insensible  tenuity,  which  move  with  amazing 
velocity,  and  only  pass  by  each  other  in  this  case  at  immense  in- 
tervals, by  which  they  are  separated  from  each  other, 

4.  It  is  here  proper  to  speak  of  certain  phenomena  which  seem, 
at  first  sight,  to  be  opposed  to  what  we  have  laid  down  with  regard 
to  the  impenetrability  of  matter,  but  which,  examined  more  atten- 
tively, only  tend  to  confirm  it. 

When  a  solid  body  is  suffered  to  fall  into  any  fluid,  as  water  for 
example,  it  sinks  and  seems  to  penetrate  the  fluid  ;  but  it  in  fact 
only  separates  and  displaces  the  parts  that  compose  it ;  for  if  the  ves- 
sel containing  the  fluid  be  formed  with  a  narrow  neck  toward  the 
top,  like  a  bottle,  the  fluid  will  be  seen  to  rise  as  the  body  enters, 
and  to  a  greater  or  less  height,  exactly  in  proportion  to  the  size  of 
the  immersed  body.  What  has  taken  place  therefore,  is  only  a  di- 
vision and  separation  of  parts,  and  not  strictly  a  penetration.  The 
same  may  be  said  when  an  edged  tool  is  forced  into  a  block  of 
wood,  only  the  parts  of  the  wood  are  separated  with  more  difficulty 
than  those  of  water.  The  same  may  be  said  also  when  a  nail  is 
driven  into  clay,  lead,  or  gold,  in  which  cases  it  only  makes  an  open- 
ing sufficient  for  its  admission.  Indeed  the  mass  thus  pierced  is  not 
entirely   separated,   but  the   parts   are    nevertheless   pressed   and 


Preliminary  Remarks.  3 

crowded  together ;  and  if  we  examine  those  which  surround  the 
opening,  caused  by  the  nail,  we  shall  find  sensible  marks  of  this 
pressure.  The  nail  in  its  turn  may  likewise  be  pierced  by  steel, 
and  this  again  by  other  bodies. 

We  hence  infer  that  bodies,  even  the  most  hard  and  compact, 
are  not  composed  of  matter  absolutely  continuous,  but  of  parts  ag- 
gregated together,  and  placed  at  distances,  which,  under  the  influ- 
ence of  external  causes,  may  become  greater  or  less.  It  is  on 
this  account  that  the  dimensions  of  any  given  mass  of  matter  are 
capable  of  being  increased  by  heat,  or  diminished  by  cold,  that  the 
panicles  of  salt  admit  of  being  separated  and  distributed,  and  as  it 
were,  lost  among  the  particles  of  water;  that  mercury  attaches 
itself  to  a  piece  of  gold  immersed  in  it,  and  insinuates  itself  into 
the  interior  of  this  compact  substance.  These  mixtures  and  disso- 
lutions sometimes  take  place  without  any  apparent  augmentation  of 
bulk,  this  bulk  being  estimated  according  to  the  exterior  surface  of 
the  bodies  in  question,  without  regard  being  had  to  the  void  spaces, 
sensible  or  insensible  to  us,  which  may  be  found  to  exist  among 
their  parts.  In  all  this  there  is  only  separation  and  mixture  without 
any  actual  penetration  of  material  particles. 

This  want  of  material  continuity  in  bodies  is  known  under  the 
general  name  of  porosity,  and  we  call  jyores  the  interstices  or  empty 
spaces  by  which  these  particles  are  separated  from  each  other. 
Porosity  seems  to  be  a  property  common  to  all  bodies,  although  it 
does  not  belong  to  the  essence  of  matter,  since  we  can  conceive  of 
sensible  bodies  which  are  entirely  destitute  of  void  space. 

5.  Thus  admitting  that  bodies  may  be  considered  as  composed 
of  smaller  parts  which  constitute  their  essence,  we  may  be  asked, 
what  is  the  form  and  magnitude  of  these  parts.  As  to  the  magni- 
tude, it  should  seem  that  it  is  extremely  minute;  for  to  whatever  ex- 
tent we  carry  the  division,  in  the  case  of  gold,  for  example,  by  the 
processes  of  wire-drawing,  filing,  and  beating,  the  smallest  particles 
preserve  invariably  all  the  properties  that  belong  to  the  entire  mass. 
Crystallized  bodies  reduced  to  an  almost  impalpable  powder,  upon 
being  examined  with  a  microscope,  are  found  to  exhibit  the  same 
forms  and  the  same  angles  which  characterize  the  whole  mass  of 
the  crystal.  We  have  examples  of  a  division  carried  to  a  still 
greater  extent  in  odors,  the  sense  being  affected  in  this  case  by  par- 
ticles proceeding  from  the  odoriferous  body  that  are  absolutely  in- 


4  Preliminary  ftemarks. 

visible  and  impalpable.  From  these  few  instances,  and  a  thousand 
others  that  might  be  mentioned,  it  is  evident  that  a  body  without 
changing  its  character,  without  ceasing  to  be  of  the  same  identical 
nature  with  the  largest  masses  that  surround  us,  may  be  divided 
into  parts,  the  smallness  of  which  eludes  the  power  of  the  senses, 
and  almost  that  of  the  imagination. 

6.  The  question  has  been  much  discussed,  whether  matter  be 
infinitely  divisible  ;  but  it  is  now  pretty  generally  agreed  that  the 
dispute  is  about  words.  If  the  point  in  question  relate  to  abstract 
geometrical  divisibility,  there  can  be  no  doubt  of  the  truth  of  the 
affirmative ;  for  however  infinitely  small  we  suppose  a  particle,  from 
the  very  circumstance  of  its  being  extended,  we  can  always  con- 
ceive this  extent  divided  into  two  halves,  and  each  of  these  into  two 
others,  and  so  on  without  end  (Calc.  4).  But  if  we  mean  by  the 
question  an  actual  physical  divisibility,  nothing  can  be  decided  ab- 
solutely one  way  or  the  other.  It  seems  however  by  all  we  can 
learn,  that  we  should  at  some  stage  of  the  division  arrive  at  materi- 
al particles  which  would  not  admit  of  being  broken,  or  altered,  or 
transmuted  the  one  into  the  other ;  for  to  whatever  chemical  opera- 
tion they  are  subjected,  into  whatever  combinations  they  are  made 
to  enter,  however  they  may  be  brought  to  constitute  a  part  of  living 
beings,  they  always  return  to  their  former  state,  widi  their  original 
properties  unchanged.  The  infinite  variety  of  processes  of  diis 
kind  through  which  the  same  material  particles  have  been  made  to 
pass  since  the  world  was  created,  does  not  appear  to  have  produced 
the  smallest  alteration. 

7.  But  how  can  such  a  system  of  particles  exist  collected  to- 
gether in  the  form  of  solid  and  resisting  masses,  as  we  see  they  are 
in  a  great  number  of  bodies,  in  all  indeed  when  they  are  properly 
examined  ?  This  state,  as  we  shall  see  hereafter,  is  produced  and 
maintained  by  the  natural  powers  with  which  all  parts  of  matter  are 
endued,  and  which  cause  them  to  tend  toward  each  other,  as  it  were 
by  an  attraction.  But  if  there  existed  only  forces  of  this  kind,  the 
particles  would  continue  to  approach  till  they  came  into  actual  con- 
tact with  each  other,  that  is,  until  they  were  arrested  by  the  impen- 
etrability of  their  parts,  which  would  not  admit  of  the  contraction 
and  dilation  which  are  constantly  observed  in  bodies.  We  accord- 
ingly infer  that  there  is  a  general  cause  of  interior  repulsion  in  bod- 
ies, by  which  the  attractive  forces  are  continually  balanced.      This 


Preliminary  Remarks.  5 

cause,  which  resides  in  all  bodies,  seems  to  be  referable  to  the  prin- 
ciple of  heat.  The  particles  of  each  body,  actuated  at  the  same 
time  by  these  two  opposite  forces,  naturally  put  themselves  in  a 
state  of  equilibrium,  resuking  from  a  compensation  of  energies,  and 
they  approach  and  recede,  according  as  the  forces  to  which  they  are 
exposed  from  without,  favor  the  attractive  or  repulsive  principle.  It 
is  with  these  minute  bodies  as  it  is  with  the  planets  of  our  system, 
which  are  found  to  move  and  oscillate,  as  it  were,  in  orbits  of  varia- 
ble forms  and  dimensions,  without  the  system  being  destroyed,  or 
the  general  equilibrium  being  disturbed.  From  these  different  con- 
ditions of  equilibrium  arise,  as  we  shall  see  hereafter,  all  the  second- 
ary and  changeable  forms  of  bodies,  such,  for  example,  as  are 
denominated  aeriform,  liquid,  solid,  crystallized,  hard,  elastic,  &c. 

8.  In  all  the  phenomena  which  present  themselves,  the  particles 
of  matter  act,  or  rather  are  acted  upon,  as  if  they  were  perfectly 
inert,  that  is,  deprived  of  all  power  of  self-direction.  They  can  be 
moved,  displaced,  stopped,  by  causes  foreign  to  themselves  ;  but  we 
never  have  been  able  to  discover  the  least  trace  of  any  thing  like 
choice  or  will,  proper  to  the  particles  themselves.  If  the  ball  which 
rolls  upon  a  billiard-table,  in  consequence  of  the  impulse  that  is  given 
to  it,  loses  by  litde  and  little  its  velocity,  and  at  length  comes  to  a 
state  of  rest,  it  is  entirely  the  effect  of  the  continual  resistance  that 
it  meets  with  from  the  roughness  of  the  cloth  with  which  it  comes 
in  contact,  and  from  the  particles  of  the  air  through  which  it  passes. 
JNIake  the  cloth  more  smooth,  or  the  air  more  jare,  and  the  same 
impulse  would  keep  it  longer  in  motion  ;  substitute  for  the  cloth  a 
marble  slab  highly  polished,  or  a  band  of  stretched  wire,  the  elas- 
ticity of  which  is  still  more  perfect,  and  the  ball  would  continue  its 
motion  for  a  much  longer  time  ;  from  all  which  it  is  to  be  inferred, 
that  if  the  obstacles  were  completely  removed  there  would  be  no 
diminution  of  the  velocity  first  communicated,  and  the  motion  would 
never  cease.  A  stone  thrown  from  the  top  of  a  tower,  and  urged 
at  die  same  time  by  the  impulse  of  the  hand  and  by  gravity,  will 
come  to  the  ground  after  proceeding  a  certain  distance,  losing  at  the 
same  time  its  horizontal  velocity,  by  imparling  it  to  the  particles  of 
air  against  which  it  impinges.  But  let  us  suppose  the  air  removed 
or  annihilated,  and  the  projectile  force  (in  the  direction  of  a  tangent), 
to  be  sufficient  to  carry  the  stone  as  far  from  the  earth  as  gravity  would 
cause  it  to  descend  each  instant,  and  the  stone  would  describe  a  circle 


6  Preliminary  Remarks. 

round  the  earth,  and  if  there  were  nothing  to  stop  or  obstruct  it,  it  would 
thus  continue  to  revolve  without  end.    We  have  indeed  this  principle 
exemplified  in  the  motion  of  the  moon,   which  revolves  in   a  void 
about  the  earth  ;  we  see  moreover  the  same  renewed,   perpetual 
motion  in  the  planets,  which  pass  in  like  manner  through  spaces  des- 
titute of  all  material  resistance.     We  are  hence  lead  to  believe  that 
matter  is  incapable  of   effecting    any    change    in   itself,  either   with 
respect  to  motion  or  rest ;  and,  once  put  into  either  of  these  states,  it 
would  continue  in  this  state  so  long  as  it  should   remain  undisturbed 
by  any  cause  foreign  to  itself.     This  indifference  to  motion  and  rest, 
this  want  of  all  power  of  self-direction,  has  obtained  the   name  of 
inertia.     There  is  one  class  of  bodies,  however,  that  seems  to  form 
an  exception  to  this  law  of  matter.     It   comprehends  those   which 
we  call  animated,  which  put  themselves  in  motion  or  stop  themselves 
by  an  act  of  the  will ;  but  even  in  these  the  material  elements  which 
constitute  their  parts  or  members,  and  these  members  themselves, 
are  perfectly  inert.     It  is  their  union  or  combination  that   possesses 
the   quality  of  life.     Separated,  they    have  no  longer  this  power, 
but  return  to  the  condition  of  ordinary  matter.     We  are    entirely  in 
the  dark  with  regard  to  the   cause  of  this  remarkable  difference  in 
the  bodies  that  surround  us.     As  to  what  constitutes  a  state  of  life, 
we   can  pretend  to   no  knowledge    whatever.      But   seeing    matter 
under  all  other  circumstances  destitute  of  the   power  of   self-direc- 
tion, and  knowing  also  that  in  living  beings   it  loses  this   faculty  by 
death  and  by  sleep,  we  are  led  to  regard  it  as  foreign  to  the  essence 
of  matter,  and  to  qpnsider  the  volition  of  animated  beings,   as  the 
act  of  an  immaterial  principle  which  resides  within  them.     We  are 
unable  to  say  in  what  part  this  principle  is  seated,  or  in  what  it  con- 
sists, and  still  less  how,  being   immaterial,   it   is   capable    of   acting 
upon  matter ;  but  with  the  litde  attention  that  we  have  paid    to  our- 
selves and  to  the  objects  about  us,  these  obscurities,  unfortunately  too 
common,  in  which  our  imperfect  knowledge  has  left  us,   ought   not 
to  be  made  the  grounds  of  an  objection  against  the  essence  of  things, 
with  which  we  must  be  contented  to  remain  unacquainted.     So  that 
we  here  proceed  philosophically,  according  to  the  rule   adopted  in 
other  cases,  by   bringing  together  things  that  are  analogous,   and 
making  the  motion  of  animated  beings  to  depend  upon  a  cause  for- 
eign to  matter ;  matter  being   found  inert  under   all  other  circum- 
stances in  which  we  have  been  able  to  examine    it.      Another  rea- 
son is  given  in  the  schools  of  philosophy  for  attributing  spontaneous 


Preliminary  Remarks.  7 

motion  to  an  immaterial  principle ;  namely,  that  the  will,  by  the 
very  nature  of  its  acts,  can  proceed  only  from  a  simple  being,  and 
that  consequently  it  cannot  belong  to  a  substance  essentially  com- 
pounded, or  at  least  divisible  and  decomposable,  like  matter  ;  but 
this  metaphysical  argument  would  carry  us  too  far  from  our  subject. 
We  content  ourselves  with  merely  suggesting  it ;  for  all  experimen- 
tal purposes,  it  will  be  sufficient  to  consider  the  irhmateriality  of  the 
principle  of  volition  as  a  distinction  founded  upon  analogy,  and  the 
inertia  of  matter  as  a  general  property  in  the  actual  state  of  the 
world. 

9.  We  are  moreover  made  acquainted  by  experiment,  with 
several  other  properties  of  matter  which  are  also  accidental,  that  is, 
which  seem  not  to  be  absolutely  necessary  in  order  that  material 
bodies  may  manifest  themselves  to  our  senses,  but  the  co-existence 
of  which  with  the  primitive  conditions  of  materiality  is  important  to  be 
known,  since  it  supplies  the  want  of  other  evidence,  in  a  great  num- 
ber of  cases  in  which  the  essential  properties  do  not  admit  of  being 
recognised.  Such,  for  example,  is  gravity.  Among  natural  bodies 
which  we  can  see  and  touch,  none  is  to  be  found  which  is  not  heavy, 
that  is,  which  does  not  tend  to  fall  toward  the  centre  of  the  eardi  when 
left  to  itself;  and  since  these  properties  are  always  found  to  accompa- 
ny each  other,  the  presence  of  the  one  is  with  respect  to  us,  always  a 
sufficient  ground  to  infer  the  existence  of  the  other.  Thus,  although 
we  can  neither  see  nor  touch  the  air,  as  we  can  see  and  touch  other 
bodies,  still  we  believe  it  to  be  a  material  substance,  because  it  is 
heavy,  capable  of  being  confined  in  vessels  and  of  exhibiting  other 
phenomena,  all  similar  to  those  which  belong  to  a  heavy  fluid.  A 
careful  examination  of  these  properties  teaches  us  at  lengdi  that 
there  are  airs  of  very  different  kinds,  which  are  all  so  many  sub- 
stances differing  essentially  from  each  other  in  the  action  which  they 
are  capable  of  exerting  on  other  bodies,  and  which  is  exerted  in 
turn  upon  them  by  these  bodies. 

10.  Moreover  attraction  is  one  of  those  contingent  properties 
which  supply  what  is  wanting  in  the  evidence  furnished  by  the  im- 
mediate testimony  of  the  senses.  I  have  said  that  the  particles  of 
all  known  bodies  exert  upon  one  another  attractive  and  repulsive 
forces  On  the  other  hand,  when  we  can  demonstrate  the  existence 
of  these  forces  in  an  unknown  principle,  we  infer  that  this  principle 
is  material.     Thus,  light  is  not  tangible  ;  it  is  not,  so  far  as  we  can 


8  Preliminary  Remarks. 

perceive,  extended  ;  it  has  no  weight,  or  at  least  none  capable  of 
being  appreciated  by  our  balances.  It  is  so  subtile  as  to  elude  all 
the  ordinary  methods  by  which  matter  manifests  itself  to  the  senses. 
But  by  causing  it  to  pass  through  transparent  bodies,  as  glass, 
water,  &.C.,  it  deviates  from  a  direct  course  in  its  passage,  and  is 
bent  precisely  as  if  it  were  repelled  by  a  force  proceeding  from  the 
surface,  and  attracted  on  the  other  hand  within  by  the  particles 
which  compose  the  transparent  body.  We  know  also  that  it  em- 
ploys a  certain  time,  very  short  indeed,  but  yet  capable  of  being 
estimated,  in  passing  from  luminous  bodies  to  us.  In  fine,  by 
subjecting  rays  of  light  to  certain  tests,  we  find  that  transparent 
bodies  attract  and  repel  them  differently  on  certain  sides  from  what 
they  do  on  others.  From  these  properties,  taken  together,  we  are 
led  to  conclude  that  light  is  a  material  substance,  composed  of 
particles  extremely  small,  the  form  of  which  is  symmetrical  on 
certain  faces,  which  are  susceptible  of  particular  attractions  and 
repulsions,  and  which  move  in  free  space,  and  through  transparent 
bodies,  with  a  given  and  determinable  velocity. 

11.  There  are  still  other  principles  which  act  upon  material 
bodies  without  being  either  visible  or  tangible,  or  susceptible  of 
being  weighed  by  our  balances,  which  even  present  much  fewer 
indications  of  materiality  than  light,  and  which  notwithstanding  are 
believed  to  be  material  substances.  Such  is  the  unknown  principle 
of  electricity.  Nothing  absolutely  material  has  yet  been  detected 
in  the  cause  of  electrical  phenomena,  nothing  indeed  which  does 
not  admit  of  being  explained  without  the  supposition  of  matter. 
Slill  in  its  distribution  over  bodies,  in  its  passage  from  one  to  the 
other  through  the  obstacles  which  separate  them,  this  principle  acts 
in  a  manner  so  exactly  conformable  to  the  laws  of  equilibrium  and 
motion  which  belong  to  fluid  substances,  that  we  can  on  this  hy- 
pothesis calculate  with  the  utmost  precision,  and  in  all  their  details, 
the  phenomena  that  are  to  take  place  under  given  circumstances. 
It  seems  extremely  probable,  therefore,  that  the  principle  in  ques- 
tion is  a  fluid,  and  that  it  is  accordingly  material.  The  same 
reasoning  is  applicable  to  the  principle  of  magnetism,  which  mani- 
fests itself  in  several  metals. 

12.  We  have  still  less  evidence  of  any  thing  material  in  the  prin- 
ciple o{  heat.  Not  only  does  it  want,  like  the  preceding,  the  sen- 
sible properties  by  which  matter  is  characterized,  but  the  laws  of 


Preliminary  Remarks  and  Definitions.  9 

its  motion  and  equilibrium  not  being  completely  known,  we  can- 
not arriv'e  at  the  same  probable  conclusion  in  this  case  as  in  the 
former.  By  following  it  however  in  our  experiments,  we  find  that 
it  diffuses  itself  in  bodies,  passes  from  one  to  another,  modifies  the 
disposition,  the  distances,  and  attractive  properties  of  their  particles. 
But  all  this  does  not  prove  inconteslably,  that  the  principle  in  ques- 
tion is  itself  a  body.  The  strongest  argument  in  favor  of  its  ma- 
teriality is  derived  perhaps  from  certain  analogies,  lately  discovered, 
between  the  radiant  properties  of  heat  and  those  of  light,  which 
lead  us  to  believe  that  one  of  these  principles  may  change  itself 
gradually  into  the  other,  that  is,  they  may  acquire  and  lose  suc- 
cessively the  modifications  by  which  they  are  respectively  distin- 
guished. The  developement  of  these  analogies  furnishes  a  most 
important  subject  of  investigation. 

13.  It  will  be  perceived  from  what  has  been  said,  that  all  bodies 
of  a  sensible  magnitude,  the  materiality  of  which  can  be  immediate- 
ly determined,  consist  in  the  grouping  together  of  a  multitude  of 
material  particles  of  extreme  minuteness,  in  which  a  difference  in 
the  mode  of  aggregation  is  the  only  circumstance  that  constitutes  a 
body  solid,  liquid,  or  gaseous.  There  are  moreover  strong  reasons 
for  believing,  as  we  have  seen,  that  these  particles  are  inert  masses, 
incapable,  from  any  inherent  power  of  their  own,  of  modifying  them- 
selves, and  susceptible  only  of  obeying  causes  from  without ;  whether 
this  want  of  choice  and  self-direction  is,  in  fact,  as  observation 
seems  to  prove,  a  general  and  essential  characteristic  of  matter,  or 
whether  we  so  regard  it  intellectually  for  the  purpose  merely  of  con- 
sidering by  themselves  those  properties  which  remain  to  matter, 
after  it  is  deprived  of  this.  Now,  material  particles  being  considered 
as  in  this  inert  slate,  there  will  hence  arise,  in  the  phenomena 
which  their  aggregation  presents,  certain  necessary  conditions, 
which  are  applicable  to  all  bodies,  independently  of  the  chemical 
nature  of  their  constituent  parts,  being  the  simple  consequences 
of  their  materiality.  Such  are  the  general  laws  of  equilibrium 
and  motion,  which  are  deduced  indeed  mathematically  from  the 
single  property  of  inertia. 

14.  We  have  already  used  the  words  rest,  motion,  and  force, 
as  making  a  part  of  ordinary  language.  It  now  becomes  necessary 
to  fix  their  meaning  with  precision.  We  begin  with  defining  the 
place  in  which  the  phenomena  under  consideradon  are  supposed  to 

Mech.  2 


10  Preliminary  Remarks  and  Definitions. 

occur.  In  order  to  this,  let  us  conceive  of  space  without  bounds, 
immaterial,  immovable,  and  of  which  all  the  parts,  similar  among 
themselves,  are  capable  of  being  penetrated  by  matter  without  op- 
posing the  smallest  resistance.  Whether  space  in  this  sense  exist  in 
nature  or  not,  is  of  little  consequence ;  the  definition  presents  to  us 
merely  an  abstract  extension.  Now  imagine  in  this  space  the  par- 
ticles of  which  we  have  been  speaking,  the  material  elements  of 
bodies  ;  and  let  us  first  consider  with  respect  to  them  the  mere  cir- 
cumstance of  their  existence.  This  simple  fact  will  be  capable  of 
two  distinct  modifications  ;  it  may  be  that  the  same  particle  shall 
remain  without  change  in  its  actual  place,  or  that  by  the  influence 
of  external  causes,  it  shall  leave  its  place  to  pass  to  some  other  part 
of  space.  The  first  of  these  states  constitutes  absolute  rest,  and  the 
second  motion. 

15.  But  we  can  conceive  further,  that  two  or  several  particles 
are  displaced  at  the  same  time,  and  impressed  with  a  common  mo- 
tion, preserving  with  regard  to  one  another  their  respective  posi- 
tions. Then,  if  we  consider  them  with  reference  to  immovable 
space,  they  will  actually  be  in  absolute  motion  ;  but  if  we  consider 
them  simply  in  their  mutual  relations  to  one  another,  these  will  con- 
tinue the  same  as  if  the  whole  group  had  remained  at  rest;  and  if 
there  were  upon  one  of  these  particles  an  intelligent  being  who 
should  observe  all  the  others,  it  would  be  impossible  for  him  to  de- 
cide from  this  observation  alone,  whether  the  whole  system  were 
in  motion  or  not.  The  permanence  of  these  relations  in  the  midst 
of  a  common  motion,  is  what  we  understand  by  relative  rest. 
This  will  be  the  condition  of  a  number  of  bodies  placed  in  a  boat  and 
abandoned  to  the  course  of  a  smooth  stream.  This  is  indeed  the 
condition  of  all  the  bodies  about  us,  so  long  as  they  remain  fixed  to 
the  same  point  of  the  terrestrial  surface.  They  are  at  rest  among 
themselves  ;  but  the  earth,  which  turns  daily  on  its  axis,  impresses 
upon  them  a  common  motion  and  at  the  same  time  bears  them  all 
together  in  its  orbit  round  the  sun,  which  perhaps  in  its  turn  carries 
the  Qprth  and  the  whole  system  of  planets  toward  some  distant 
constellation.  Relative  rest,  therefore,  is  really  the  only  kind  of  rest 
which  can  actually  take  place  among  the  objects  to  which  our  attention 
is  directed.   It  is  at  least  all  that  we  can  ever  be  certain  of  observing. 

16,  We  are  hence  led  to  make  a  similar  distinction  with  respect 
to  motion,  and  to  separate  the  absolute  motions  of  bodies,  considered 


Preliminary  Remarks  and  Definitions.  11 

with  reference  to  immovable  space,  from  the  relative  changes  of 
position  which  may  happen  among  them.  These  last  therefore  may- 
be called  rdative  motions,  whether  that  body  of  the  system  to  which 
they  are  referred,  be  itself  in  motion  or  at  rest.  The  changes  of 
place,  for  example,  among  the  heavenly  bodies,  which  we  observe 
from  the  surface  of  the  earth,  are  not  absolute  but  relative  motions, 
because  the  earth  to  which  we  refer  them,  as  a  fixed  centre,  has 
actually  a  motion  of  rotation  on  its  axis  and  a  progressive  motion 
about  the  sun.  Even  when  by  calculation  we  have  inferred  from 
these  observations  the  actual  motions  of  the  heavenly  bodies  as  they 
would  appear,  if  seen  from  ihe  sun,  we  cannot  affirm  positively  that 
these  are  absolute  motions,  since  it  may  be  that  the  sun  and  the 
whole  planetary  system  have  a  common  motion  in  space. 

17.  According  to  the  idea  of  inertia  which  we  derive  from 
experience,  we  must  regard  the  state  of  motion  and  that  of  rest, 
as  simple  accidents  of  matter,  which  it  is  incapable  of  imparting  to 
itself,  and  which  it  can  only  receive  from  without,  and  which,  once 
received,  it  cannot  alter.  When  therefore  we  see  a  body  passing 
from  one  of  these  states  to  the  other,  we  must  regard  this  change 
as  produced  and  determined  by  the  action  of  external  causes. 
These  causes,  whatever  they  may  be,  are  denominated  forces. 
Nature  presents  us  with  an  infinite  number  of  them  which  are  at 
least  in  appearance  of  different  kinds.  Such  are  the  forces  pro- 
duced by  the  muscles  and  organs  of  living  animals,  the  exercise  of 
which,  for  the  most  part,  depends  solely  on  the  will.  Such  are  also 
the  forces  of  physical  agents,  as  the  expansion  of  bodies  by  heat, 
and  their  contraction  by  cold,  he.  There  are  moreover  others 
which  seem  to  be  inherent  in  certain  bodies,  as  the  attraction  of  the 
magnet  for  iron,  and  that  which  is  manifested  among  electrified 
bodies. 

From  the  very  nature  of  matter  as  thus  presented  to  our  con- 
sideration, it  will  be  seen  that  a  body  once  put  into  a  state  of  mo- 
tion or  rest,  by  any  cause  it/iatcver,  must  continue  in  that  state  for 
ever,  if  no  new  cause  is  made  to  act  vpon  it.  If  it  cannot  give  itself 
motion  when  at  rest,  it  cannot  stop  itself  when  in  motion,  for  this 
would  be  equivalent  to  giving  itself  motion  in  the  opposite  direction  ; 
neither  can  it  change  its  velocity  or  direction,  for  this  would  equally 
imply  a  new  force.  Thus,  motion  is  naturally  equal  or  nniform 
and  rectilinear. 


12  Preliminary  Remarks  and  Definitions. 

18.  When  several  forces  are  applied  at  the  same  lime  to  a  bodyj 
they  are  mutually  modified  by  the  connexion  which  exists  among 
the  different  parts  of  tlie  body,  and  which  prevents  each  from  taking 
the  motion  which  the  force  exerted  upon  it  tends  to  produce.  If 
these  forces  happen  entirely  to  destroy  each  other,  so  that  the  body 
remains  at  rest,  we  say  that  the  forces  are  in  equilibrium,  or  that 
the  body  is  in  equilibrium,  under  the  action  of  these  forces. 

19.  Mechanics  is  the  science  which  treats  of  the  equilibrium 
and  motion  of  bodies.  That  part,  the  object  of  which  is  to  dis- 
cover the  conditions  of  equilibrium,  is  called  statics.*  We  give  the 
name  of  dynamics  f  to  the  other  part,  which  has  for  its  object  to 
determine  the  motion  which  a  body  takes,  when  the  forces  applied 
to  it  are  not  in  equilibrium.  The  general  laws  of  statics  and  dy- 
namics are  applicable  to  fluids  ;  but,  on  account  of  the  peculiar  diffi- 
culty attending  the  consideration  of  this  class  of  bodies,  we  are  ac- 
customed to  treat  them  separately.  That  part  of  the  mechanics  of 
fluids  which  relates  to  their  equilibrium  is  called  hydrostatics,  |  and 
that  which  comprehends  their  motions,  hydrodynamics.  <^ 

20.  In  our  inquiries  on  these  subjects,  we  first  proceed  upon 
the  supposition  that  there  are  no  other  bodies,  and  no  other  forces, 
in  nature,  except  those  under  consideration.  Thus  all  bodies  are 
supposed  to  be  destitute  of  weight,  and  free  from  friction,  resistance, 
and  obstructions  of  every  kind.  Regard  is  afterwards  had  to  these 
causes  ;  but  to  estimate  their  effects,  it  is  necessary  to  begin  by  in- 
vestigating each  point  separately. 

*  From  'iaTr,ixi,  I  stand.  f  From  divu^iq,  power. 

t  From  v8oiQ,  water,  and  'ioTi]iJ.L.       §  From  vdwq  and  dvvu^iig. 


STATICS. 


Chd'    ■! 

Of  Uniform  Motion, 

21.  A  body  is  said  to  have  a  uniform  motion  when  it  passes 
continually  over  the  same  space  in  the  same  time. 

In  order  to  compare   the   motions  of   two  bodies  which  move 
uniformly,  it  is   necessary  to  consider  the   space  which  each   de- 
scribes in  the  same  determinate  time,  as  one  minute,  one  second,  &g 
This  space  is  what  is  called  the  velocity  of  the  body. 

22.  The  velocity  of  a  body  therefore  is,  properly  speaking,, 
only  the  space  which  this  body  is  capable  of  describing  uniformly  in 
the  interval  of  time  which  we  take  for  unity. 

Thus  in  the  uniform  motion  of  two  bodies,  the  time  being  reck- 
oned in  seconds,  if  one  passes  over  five  feet  in  a  second,  and  the 
other  six  feet  in  a  second,  we  say  that  the  velocity  of  the  first  is 
five  feet,  and  that  of  the  second  six  feet. 

23.  But  if,  the  second  being  always  taken  as  the  unit  of  time, 
I  am  told  that  a  body  passes  over  100  feet  in  5  seconds,  100  feet 
does  not  express  the  velocity,  since  this  space  is  not  that  which 
answers  to  the  unit  of  time,  a  second  ;  but  it  will  be  perceived,  that 
in  each  second  it  would  pass  over  a  fifth  part  of  this  100  feet,  or  20 
feet ;  that  is,  in  order  to  find  the  velocity,  I  divide  the  number  100, 
the  parts  of  the  space  passed  over,  by  5,  the  number  of  units  in  the 
elapsed  time.  Hence  universally,  the  velocity  is  equal  to  the  space 
divided  by  the  time  ;  for  it  is  clear,  that  if  we  divide  the  whole  space 
into  as  many  equal  parts,  as  there  are  units  in  the  time  elapsed, 
each  part  will  be  the  space  described  during  this  unit  of  time,  and 
will  consequently  be  the  velocity  according  to  our  definition.     Thus 


14  Statics. 

calling  V  the  velocity  and  5  the  space  passed  over  in  any  portion  of 
time  denoted  by  t,  we  shall  have 

s 

this  is  one  of  the  fundamental  principles  of  mechanics. 

24.  The  equation  v  =z  —  gives   not  only  the   measure  of  the 

velocity,  but  also  that  of  the  time  and  space.  Indeed  if  we  con- 
sider t  and  s  as  unknown  quantities  successively,  we  shall  have, 
by  the  common  rules  of  algebra, 

t  =  ± 

V 

and  , 

s  ^  V  t ; 

Thus,  to  find  the  time,  we  divide  the  space  by  the  velocity ;  and,  to 
find  the  space,  we  multiply  the  velocity  by  the  time. 

If,  for  example,  it  is  asked  what  time  is  required  to  describe 
200  feet,  when  the  body  in  question  has  a  uniform  velocity  of  5 
feet  in  a  second  ;  it  is  evident  that  it  would  require  as  many  seconds 
as  there  are  5  feet  in  200  feet;  that  is,  we  should  have  the  time 
sought,  or  the  number  of  seconds,  by  dividing  the  space  200  by 
the  velocity  5  ;  we  shall  find  for  the  answer  40  seconds  ;  or,  in 
other  words,  a  number  of  seconds  equal  to  the  quotient  arising  from 
dividing  the  space  by  the  time. 

In  like  manner,  if  it  is  asked  what  space  would  be  described  in 
20  seconds  by  a  body  moving  with  a  constant  velocity  of  5  feet  in 
a  second  ;  it  is  manifest  that  it  would  describe  20  times  five  feet  j 
that  is,  it  is  necessary  in  this  case  to  multiply  the  velocity  by  the 
time. 

Thus,  although  we  have  here  employed  algebraic  characters,  it 
is  not  because  they  are  necessary  to  the  investigation  of  these  fun- 
damental truths,  but  because,  by  means  of  them,  the  propositions, 
and  their  dependence,  the  one  upon  the  other,  are  more  concisely 
expressed,  and  more  easily  remembered.  Indeed  it  will  be  seen 
by  the  above  example,  that  the  first  principle,  expressed  algebraic- 
ally, being  once  fixed  in  the  mind,  the  two  others  are  readily  de- 
duced from  it  by  the  most  familiar  rules. 

25.  It  will  be  easy  now  to  compare  the  uniform  motions  of  two, 
or  of  a  greater  number  of  bodies.     If  it  is  asked,  for  example, 


Uniform  Motion.  15 

what  is  the  ratio  of  the  velocities  of  two  bodies  which  describe  the 
known  spaces  s,  s',  in  the  times  t,  t'  respectively  ;  by  calling  v,  v', 
the  velocities  of  these  two  bodies  respectively,  we  shall  have 

v  =  j,    and    v'  =  — , 

whence 

s       s' 

that  is,  the  velocities  are  as  the  spaces  divided  by  the  times. 

In  a  word,  if  it  is  proposed  to  compare  the  velocities,  the  spaces, 
or  the  times,  ilie  principle  above  laid  down,  will  give  the  expression 
for  each  of  these  particulars  with  respect  to  each  body  ;  we  have 
therefore  only  to  compare  together  these  expressions.  For  ex- 
ample, if   we  would  compare  the  spaces,  the  fundamental  propo- 

sition  t;  =  — ,  gives  s  =:  v  t ;  we  have  in  like  manner  for  the  sec- 
ond body  s'  =1  v^  i' ;  whence 

s  :  s'  :  :  V  t  :  v'  i', 
that  is,  the  spaces  are  as  the  velocities  multiplied  by  the  times. 

26.  Of  these  three  things,  namely,  the  space,  time,  and  velocity, 
if  we  would  compare  two  together,  when  the  third  is  the  same  for 
each  body,  we  have  only  to  deduce  from  the  same  fundamental 
theorem,  the  expression  for  this  third  particular,  with  respect  to 
each  body,  and  to  put  these  two  expressions  equal  to  each  other. 
If,  for  example,  we  would  know  the  ratio  of  tiie  spaces  when  the 
velocities  are  the  same,  we  should  have 

V  =  —,  and  v'  =2  —r ;  23. 

whence,   since  by  supposition  v  =z  v',    we   have  —=z—,  and    ac- 
cordingly 

s  :  s'  :  :  t  :  t'  ; 
that  is,  the  velocities  being  equal,  the  spaces  are  as  the  times.  It 
will  be  found  in  like  manner  that,  the  times  being  equal,  the  spaces 
are  as  the  velocities ;  and  that,  the  spaces  being  equal,  the  velocities 
must  be  inversely  as  the  times.  Indeed  we  have  in  this  last  case 
s  :=  V  t,  and  s'  =^  v'  t' ;  from  which  we  obtain,  when  s  =.  s', 

v:v'::t''.t. 


1 6  Statics. 

s 
Thus  the  single  proposition  v  =:  -—  furnishes    the  means  of  com- 
paring all  the  circumstances  of  uniform  motion. 


Of  Forces  and  the   Quantitij  of  Motion. 

27.  The  sum  of  the  material  parts  of  which  a  body  consists,  is 
called  its  mass  ;  but  in  the  use  we  shall  make  of  the  word  is  to  be 
understood  the  number  of  material  parts  of  which  the  body  is  com- 
posed. 

Force,  as  we  have  said,  is  the  cause  which  either  moves  or 
tends  to  move  a  body. 

As  forces  interest  us  only  by  their  effects,  it  is  by  the  efll^cts  of 
which  they  are  capable,  that  we  are  to  measure  them.  Now  the 
effect  of  a  force  is  to  cause  in  each  particle  of  a  body  a  certain  ve- 
locity. Accordingly,  if  all  the  parts  receive  the  same  velocity  as 
is  here  supposed,  the  effect  of  the  moving  cause  has  for  its  measure 
the  velocity  multiplied  by  the  number  of  material  parts  contained  in 
the  body,  that  is,  by  the  mass.  Therefore,  a  force  is  measured  by 
the  velocity,  which  it  is  capable  of  impressing  upon  a  known  mass, 
multiplied  by  this  mass. 

28.  The  product  of  the  mass  of  a  body  by  its  velocity  is  called 
the  quantity  of  motion  of  this  body.  Forces  are  therefore  measured 
by  the  quantities  of  motion  which  they  are  capable  of  producing  re- 
spectively. Thus,  if  we  designate  the  above  product  by^,  ihe  mass 
by  m,  and  the  velocity  by  v,  we  shall  have  jj  z=  m  v.  This  equa- 
tion eives  v  =.  —,  and  m  =  —  ;  from  which  it  will  be  seen  that, 

°  7n  V 

1.  The  moving  force  of  a  body  and  its  mass  being  known,  we 
shall  find  the  velocity  by  dividing  the  moving  force  by  the  mass  ; 

2.  The  moving  force  and  the  velocity  being  known,  we  shall  find 
the  mass  belonging  to  this  velocity  and  moving  force,  by  dividing  the 
moving  force  by  the  velocity  ; 

3.  If  the  moving  forces  are  equal,  ihe  velocities  are  inversely  as 
ihe  masses. 

The  truth  of  these  propositions  is  easily  shown  by  putting  suc- 
cessively the  value  of  m  equal  to  m',  that  of  v  equal  to  v',  and  that 


Uniform  Motion.  17 

of  p  equal  to  p' ;  the  equations  thus  obtained,  reduced  and  convert- 
ed into  proportions,  form  the  several  propositions  above  stated. 


Remark. 

29.  The  mass,  or  number  of  material  parts  of  a  body,  depends 
upon  its  bulk  or  volume,  and  vvhai  is  called  its  density,  that  is,  the 
greater  or  less  degree  of  closeness  or  proximity  among  its  particles. 
As  all  bodies  have  more  or  less  of  void  space  within  them,  their 
quantities  of  matter  are  not  proportional  to  their  bulks  ;  since,  under 
tlie  same  bulk,  tlie  quantity  of  matter  is  greater  according  as  the 
parts  are  more  crowded  and  compressed  together.  A  body  is  said 
to  be  more  dense  than  another,  when  under  the  same  bulk  it  has 
more  matter ;  and,  on  the  other  hand,  to  be  more  rare  than  an- 
other, when  under  the  same  bulk  it  has  less  matter. 

Accordingly,  by  means  of  the  density  of  a  body,  we  are  able, 
when  the  bulk  is  known,  to  judge  of  the  number  of  material  parts 
which  compose  it ;  so  that  the  density  may  be  considered  as  repre- 
senting the  number  of  material  parts  in  a  given  bulk.  When  we 
say  that  gold  is  19  times  as  dense  as  water,  we  mean  that  gold 
contains  19  times  as  many  parts  in  the  same  space. 

30.  By  considering  density  as  expressing  the  number  of  material 
parts  of  a  determinate  bulk,  taken  as  the  unit  of  bidk,  it  is  evident 
that  in  order  to  find  the  mass,  or  total  number  of  material  parts,  of  a 
body  whose  bulk  is  known,  we  should  simply  multiply  the  density 
by  the  bulk.  If,  for  example,  the  density  of  a  cubic  inch  of  gold 
be  represented  by  19,  the  quantity  of  matter  contained  in  10  cubic 
inches  would  be  10  times  19.  Thus,  designating  the  mass  by  m,  the 
bulk  by  b,  and  ilie  density  by  d,  we  sliall  have 

m  =  b  X  J>' 
It  will  hence  be  easy  to  compare  together  the  masses,  the  bulks, 
and  the  densities  of  bodies. 

Moreover,  as  the  particles  of  matter,  of  whatever  kind,  tend  by 
the  force  of  gravity  to  move  with  the  same  velocity,  or  exert  the 
same  power,  the  combined  action  arising  from  this  cause  will  be 
proportional  to  the  nuniber  of  particles;  that  is,  the  weight  of  a  body 
is  as  its  density,  other  things  being  the  same.  The  relative  weights 
of  the  different  kinds  of  matter  under  equal  bulks,  are  called  the 
Mech.  3 


1 8  Statics. 


specific  gravities  of  the  bodies  respectively,  the  weight  of  pure  water, 
in  a  vacuuin,  at  a  particular  temperature,  being  taken  as  the  unit. 
Thus,  if  a  cubic  inch,  a  cubic  foot,  &-c.,  of  gold  be  19  times  heavier 
than  the  same  bulk  of  water,  under  the  same  circumstances,  as  to 
temperature,  &-c.,  the  specific  gravity  of  gold  is  said  to  be  19. 


Of  Equilihrium  between  Forces  directly  opposite. 

31.  We  shall  represent  forces,  as  we  have  said,  by  their  effects, 
that  is,  by  the  quantities  of  motion  which  they  are  capable  of  pro- 
ducing respectively,  in  a  determinate  mass.  But,  not  to  embrace  too 
many  objects  at  once,  we  shall  consider  each  mass  or  body,  as  re- 
duced to  a  single  point,  at  which  we  suppose  the  same  quantity  of 
matter  as  in  the  body  of  which  it  takes  the  place.  We  shall  see 
hereafter  that  there  is  in  fact  in  every  body  a  point  through  which 
motion  is  transmitted,  as  if  the  whole  mass  were  concentrated  there. 
We  shall,  moreover,  unless  the  contrary  is  expressly  stated,  consider 
bodies  as  composed  of  particles  absolutely  hard,  and  connected 
together  in  such  a  manner  as  not  to  admit  of  any  change  in  their 
respective  situations  by  the  action  of  any  force  whatever. 

32.  This  being  premised,  let  us  suppose  two  bodies  m,  n,  to  be 
put  in  motion,  the  first  from  A  toward  C,  with  a  velocity  m,  the  sec- 

^'  '  ond  from  C  toward  A  with  a  velocity  v.  When  these  bodies  come 
to  meet,  they  will  be  in  equilibrium,  if  the  quantity  of  motion  in  m 
is  equal  to  the  quantity  of  motion  in  n  ;  that  is,  if  m  m  is  equal  to  n  v. 

Indeed  it  is  evident,  that  if  m  is  equal  to  n,  and  the  velocity  u  is 
equal  to  v,  there  must  be  an  equilibrium  ;  for  in  this  case,  whatever 
reason  there  may  be  for  supposing  m  to  prevail  over  n,  might  also 
be  given  for  supposing  n  to  prevail  over  m,  since  they  are  by  hypo- 
thesis in  all  respects  equal. 

33.  Let  us  suppose  now,  that  m  is  double  of  n,  but  that  v,  at  the 
same  time,  is  double  of  m,  that  is,  that  n  passes  over  two  feet,  for 
example,  in  a  second,  while  m  passes  over  one  foot  in  a  second. 
It  is  clear  that  we  may  consider  m  as  composed  of  two  masses  equal 
each  to  n  ;  and  that,  at  the  instant  of  meeting,  we  may  represent  the 
body  n  as  having  a  velocity  of  one  foot  in  a  second,  to  which  is 
added,  at  the  same  instant,  another  velocity  of  one  foot  in  a  second. 
We  may  then  conceive,  that,  in  meeting,  the  mass  n  expends  one  of 


Equilibrium  between  Forces  directly  opposite.  19 

its  velocities  against  a  portion  of  the  mass  m  equal  to  itself,  and 
its  other  velocity  against  the  remaining  portion  of  m  of  the  same 
magnitude. 

If  now,  instead  of  supposing  the  masses  m  and  n  in  the  ratio  of 
2  to  1,  and  their  velocities  in  the  ratio  of  1  to  2,  we  suppose  them 
in  any  other  ratio,  it  is  evident  that  we  may  always  conceive  the 
greater  mass  as  decomposed  into  a  certain  number  of  portions  equal 
each  to  the  smaller,  and  of  which  each  shall  destroy,  in  the  smaller, 
a  velocity  equal  to  its  own.  We  may  therefore  consider  the  fol- 
lowing proposition  as  established. 

Two  bodies  which  act  directly  against  each  other  in  the  same 
straight  line,  are  in  equilibrium  when  their  quantities  of  motion 
are  equal;  that  is,  when  the  product  of  the  mass  of  the  one 
into  the  velocity  with  which  it  moves,  or  tends  to  move,  is  equal 
to  the  product  of  the  mass  of  the  other  into  its  actual  or  virtual  ve- 
locity. 

This  proposition  is  to  be  regarded  as  general,  whether  the  two 
bodies  move  freely  and  directly  the  one  against  the  other,  or  whether 
they  act  against  each  other  by  the  intervention  of  a  rod  inflexi- 
ble and  without  mass,  or  whether  they  are  considered  as  pulling  in 
opposite  directions  by  means  of  a  thread  m  n  incapable  of  being 
extended.  And  reciprocally,  if  two  bodies  are  in  equilibrium,  we 
may  conclude  that  their  motions  are  directly  opposite,  and  that  their 
quantities  of  motion  are  equal. 

34.  We  infer,  moreover,  that  if  three  or  a  greater  number  of 
bodies  m,  n, ,  o,  &;c.,  moving,  or  tending  to  move,  in  the  same  Fig.  2, 
straight  line,  with  velocities  m,  v,  tv,  he,  are  in  equilibrium,  the 
sum  of  the  quantities  of  motion  of  those  which  act  in  one  direction 
is  equal  to  the  sum  of  the  quantities  of  motion  of  those  which  act  in 
the  opposite  direction.  For,  tliey  being  in  equilibrium,  we  may 
always  suppose  that,  m  and  n  acting  in  the  same  direction,  n  de- 
stroys a  part  of  the  motion  of  o,  and  that  m  destroys  the  remaining 
part.  Now  if  we  represent  by  x  the  velocity  that  o  loses  by  the 
action  of  n,  we  shall  have  o  x,  for  the  quantity  of  motion  destroyed 
by  the  action  of  n  ;  we  have  accordingly 

n  V  =^  0  X. 
The  body  m,  therefore,  will  have  only  to  destroy  in  a  the  remain- 


20  Statics. 

ing  quantity,  namely,  o  iv  —  ox;  we  have  consequently 

m  u  ^=.  0  w  —  ox; 
or  since  o  x  =■  n  v, 

m  u  ^=.  o  w  —  11  V, 
that  is, 

m  u  -\-  n  V  ^=-  0  w. 

/ 
Of  Compound  Motion. 

35.  We  still  consider  the  masses  to  which  the  forces  under  con- 
sideration are  applied,  as  concentrated  each  in  a  point. 

We  call  compound  motion  that  which  takes  place  in  a  body, 
when  urged  at  the  same  time  by  two  or  more  forces  having  any 
given  direction  with  each  other. 

Fig.  3.  If  a  body  m  moving  in  the  line    CB,  receive,  upon  arriving  at 

the  point  A,  an  impulse  in  the  direction  AD,  perpendicular  to  CB, 
this  impulse  can  produce  no  other  effect,  except  that  of  removing 
the  body  from  CB.  It  can  neither  augment  nor  diminish  the  velo- 
city with  which,  at  the  lime  of  receiving  the  impulse,  it  was  depart- 
ing from  AD.  Indeed,  since  AD  is  perpendicular  to  CB,  there  is 
no  reason  why  a  force  acting  in  the  direction  AD  should  produce 
an  effect  to  the  right,  rather  than  to  the  left,  of  this  line,  and  as  it  can- 
not act  in  both  these  directions  at  once,  it  can  have  no  influence 
either  way. 

The  same  reasoning  will  hold  true,  if  we  suppose  that  the  body 
m,  moving  in  the  line  AD,  receives,  upon  arriving  at  A,  an  impulse 
in  the  direction  AB.  This  impulse  will  neither  add  to  nor  take 
from  the  velocity  with  which  the  body  m  was  departing  from  AB. 

' 'g-  4-  36.  If  two  forces  p  and  q,  the  directions  of  which  are  at  right 

angles  to  each  other,  act  at  the  same  instant  upon  a  body  m,  and  the 
force  q  is  such  as  by  its  sole  and  instantaneous  action  to  cause  the  body 
to  pass  over  AB  in  a  determinate  time,  as  one  second,  and  the  force  p  is 
such  as  to  cause  the  body  to  pass  over  AD  in  the  same  time,  we  say 
that  by  the  joint  action  of  the  two  forces,  q  and  p,  the  body  m  will 
in  the  same  time  pass  over  the  diagonal  AE,  of  the  parallelogram 
DABE,  which  has  for  its  sides  these  same  lines,  AB,  AD. 

Since  the  two  forces  act  at  the  same  instant  upon  the  given  body, 
we  may  suppose  it  moving  in  the  line  AD,  and  that   at  the  in- 


Compound  Motion.  2\ 

stant  of  Its  arriving  at  the  point  A,  it  receives  the  force  5-  in  a  direc- 
tion perpendicular  to  AD.  Now,  according  to  article  35,  the  force 
q  can  neither  increase  nor  diminish  the  velocity  with  which  it  was 
at  this  moment  departing  from  AB ;  if,  therefore,  diroiigh  the  point 
D  we  draw  DE  parallel  to  AB,  the  body  must  at  the  end  of  a  sec- 
ond be  somewhere  in  the  line  DE,  all  parts  of  which  are  equally 
distant  from  AB. 

The  same  reasoning  may  be  adopted  with  regard  to  the  force  q, 
by  which  it  will  be  seen,  that  if,  through  the  point  B,  we  draw  BE 
parallel  to  AD,  the  body  must  at  the  end  of  a  second  be  some- 
where in  BE.  But  there  is  only  the  point  E  which  is  at  the  same 
time  in  DE  and  BE ;  therefore  at  the  end  of  a  second  the  body 
will  be  in  E. 

It  is  also  evident,  that  whatever  course  the  body  takes,  by  the 
instantaneous  action  of  the  forces,  this  course  must  be  a  straight 
line,  since,  from  the  instant  that  the  forces  are  exerted,  the  body  17. 
is  abandoned  lo  itself,  and  there  is  no  cause  to  incline  it  one  way 
rather  than  another.  Accordingly,  as  this  body  passes  through  A  and 
E,  and  without  any  thing  to  change  its  direction,  the  course  must 
be  AE,  that  is,  the  diagonal  of  the  parallelogram  DABE. 

We  will  add  moreover,  that  the  body  describes  AE  with  a  uni- 
form motion,  since,  after  the  joint  action  of  the  two  forces,  it  is  left 
equally  without  any  cause  to  alter  its  rate  of  moving.  ^'^• 

37.  Since  the  two  forces  j»  and  q,  acting  simultaneously  upon  the 
body  m,  have  no  other  effect  than  to  make  it  describe  the  diagonal 
AE,  we  infer,  that,  instead  of  two  forces  whose  directions  are  at  right 
angles  to  each  other,  we  may  always  substitute  a  single  one,  pro- 
vided that  this  single  one  is  such  as  to  cause  the  body  to  describe  the 
diagonal  of  a  right-angled  parallelogram,  the  sides  of  which  would 
be  described  in  the  same  time,  each  separately,  by  the  action  of  the 
force  of  which  it  represents  the  direction. 

The  single  force  AE,  which  results  from  the  action  of  the  two 
forces  AB,  AD,  is  called  the  resultant  of  these  two  forces.  As  the 
lines  AB,  AD,  represent  the  effects  which  the  forces  q  and  p  are 
singly  capable  of  producing,  and  AE  the  effect  which  they  are  able 
to  produce  conjointly,  we  may  regard  AB,  AD,  AE.  as  represent- 
ing these  forces  themselves. 

We  mny  thus  consider  any  single  force  AE,  as  being  the  result 
of  two  other  forces  AB,  AD,  the  directions  of  which  are  at  right 


22  Statics. 

angles  to  each  other,  provided  that,  the  first  being  represented  by 
the  diagonal  AE,  the  others  are  represented  by  the  sides  AB,  jiD, 
of  this  same  right-angled  parallelogram.  For  the  single  force  AE, 
therefore,  we  may  substitute  the  two  forces  AB,  AD,  since  these 
two  will  in  fact  only  produce  AE. 

33.  In  general,  whatever  be  the  angle  formed  by  the  directions 
ig.  5,6.  of  the  tivo  forces  p  and  q  which  act  at  the  same  time  upon  a  body  m, 
this  body  will  still  describe  the  diagonal  AE,  of  the  parallelogram 
DABE,  the  sides  of  which  represent,  in  the  directions  of  the  forces, 
the  effects  which  they  are  separately  capable  of  producing  ;  and  the 
body  will  describe  this  diagonal  in  the  same  time  in  ivhich,  by  the  ac- 
tion of  either  of  the  two  forces,  it  tvould  have  described  the  side 
which  represents  this  force. 

Through  the  point  A  let  the  line  FAHhe  drawn  perpendicular 
to  the  diagonal  AE,  and  through  the  points  J)  and  B,  let  JDFy 
BH,  be  drawn  parallel,  and  DG,  Bl,  perpendicular  to  the  diagonal 
AE.  Instead  of  the  force  p,  represented  by  AD,  the  diagonal  of  the 
rectangular  parallelogram  FAGD,  we  may  take  the  two  forces  AF, 

37.  AG.  For  the  same  reason,  instead  of  the  force  q,  represented  by 
the  diagonal  AB,  of  the  rectangular  parallelogram  AHBl,  we  may 
take  the  two  forces  AH,  AI.  We  may  therefore,  instead  of  the  two 
forces  p  and  q,  substitute  the  four  forces  AF,  AG,  AH,  Aly  and 
these  cannot  but  have  the  same  resultant  as  the  two  forces  p  and  q. 
Now  of  these  four  forces,  the  two  AH,  AF,  contribute  nothing  to 
the  resultant,  because  they  act  in  opposite  directions,  and  are  equal 
to  each  other.     Indeed  it  will  be  readily  seen,  that,  from  the  na- 

32.  ture  of  a  parallelogram,  the  two  triangles  DGA,  ElB,  are  equal  j 
therefore  DG  =  Bl,  and  consequently  AF  =  AH. 

As  to  the  two  forces  AI,  AG  (Jig.  5.),  since  they  are  exerted 
according  to  the  same  line,  and  are  directed  the  same  way,  the  re- 
sult must  be  the  sum  of  the  two  effects  AG,  AI ;  and  in^o-.  6,  since 
AI,  AG,  are  exerted  according  to  the  same  line,  and  are  opposed 
the  one  to  the  other,  the  result  must  be  the  difference  of  the  two 
effects  AG,  AI.  But  as  the  triangle  EIB  is  equal  to  DGA,  we 
shall  have  {fg.  5.) 

AI-{-AG  =  AI+EI  =  AE', 

and  {fig.  6.) 

A1  —  AG  =  A1  —  EI=AE. 

We  concludej  therefore,  that  the  four  forces  AF,  AH,   AG,  AI, 


Compound  Motion.  23 

and  consequently  the  two  forces  AD,  AB,  have  no  other  effect,  than 
the  force  AE,  represented  by  the  diagonal  of  the  parallelogram 
DjIBE,  of  which  the  two  sides  AB,  AD,  denote  the  forces  q,  p. 
This  proposition  is  known  by  the  name  of  the  parallelogram  of  forces. 

39.  We  have,  in  what  precedes,  represented  the   two   forces 
p,  q,  by  the  lines  AD,  AB,  which  they  are  capable  of  making  theFig.  4,5^ 
body  m  describe  in  the  same  lime,  that  is,   by  the   velocities  which 
they  would   communicate ;  although,  according   to   what   we   have 
said,  the  true  measure  of  any  force  is  the   quantity  of  motion  that    28. 
it  is  capable  of  producing.     But  as  the   quantities  of  motion   are  in 
the  ratio  of  the  velocities,  when  the  mass  is   the  same,  as  is  the  fact    29. 
in  the  present  case ;  we  may  always,  as  we   have  now   done,   take 
the  velocities  AD,  A  B,  as  representing  the  two  forces. 

But  if,  instead  of  having  immediately  the  velocities  which  the 
two  forces  p,  q,  are  capable  of  giving  to  the  body  m,  we  had  the 
quantities  of  motion  which  they  would  produce  in  known  masses, 
we  should  take  AD,  AB,  in  the  ratio  of  these  quantities  of  motion. 
If,  for  example,  I  know  the  forces^,  q,  only  by  this  circumstance, 
that  the  force  p  is  capable  of  giving  a  known  velocity  u,  to  a  known 
mass  n  ;  and  that  the  force  q  is  capable  of  giving  a  velocity  t;  to  a 
known  mass  o ;  I  should  take 

AD   :    AB    : :    nu   :   o  v. 
For,  according  to  what  has  been  shown,  AD,  AB,  are  to  be  taken 
in  the  ratio  of  the  velocities  which  they  are  capable  of  giving  to  the 
body  m.     Now  the  first  being  capable  of  producing  the   quantity  of 

motion  n  u,  is  capable  of  giving  to  the  body  m  the  velocity  — ^.     For     28. 

the  same  reason  the  second,  or  the  force  q,  is  capable  of  giving  to 

0  V 

the  body  m,  the  velocity  — .  The  lines  AD,  AB,  are  conse- 
quently  to  be  taken  according  to  the  following  proportion  ; 

AD  .AB::"^'."^; 

m        m 

,  n  u        0  V 

but  —    :    —  :  :  n  u  :  0  V. 

m  m 

We  see  therefore,  that  AD,  AB,  must  be  in  the  ratio  of  the  quan- 
tities of  motion  n  u,  o  v,  which  are  the  measures  respectively  of  the 
forces  p,  q. 


24  Statics. 

What  has  now  been  remarked,  will  be  found  useful  in  compar- 
ing the  effects  of  different  forces  applied  to  different  bodies. 

38.  The  general  proposition  above  demonstrated  is  of  the    greatest 

importance,   as    almost  every  thing  we  have   to  offer,  consists  in 
an  application  of  it. 

40.  From  what  has  been  said,  it  will  be  seen  that  it  is  imma- 
terial whether  we  regard  a  body  as  urged  by   the   combined   action 

Fig.  5,6.  of  the  two  forces  AB,  AD,  which  make  with  each  other  any  as- 
sumed angle,  or  whether  we  regard  it  as  urged  by  the  single  action 
of  a  force  represented  by  the  diagonal  AE. 

And  reciprocally,  it  amounts  to  the  same  thing,  whether  we 
consider  a  body  as  urged  by  a  single  force  AE,  or  by  two  forces 
represented  by  the  two  sides  of  a  parallelogram  of  which  the  sin- 
gle force  AE  is  the  diagonal.  Let  a  body,  for  example,  be  sup- 
posed to  pass  from  A  lo  E  by  a  uniform  motion  in  one  second,  or 
let  it  be  supposed  to  move  through  AB  at  such  a  rate  as  to  describe 
it  in  one  second,  while  in  the  same  time  this  line  is  carried  parallel 
to  itself  along  AE;  in  this  case,  as  in  the  former,  the  body  will 
merely  describe  the  line  AE. 

41.  The  two  forces  AB,  AD,  meeting  at  the  point  ./Z,  are   ne- 
Geom.   cessaiily  in  the  same  plane.      Since,  therefore,  they    have  for  their 
321-       resultant  the  diagonal  AE,  which  is  in  the  plane  of  the  parallelo- 
gram, we  may  infer  generally  that  any  two  forces  which  unite  in  the 
same  point  are  always  in  the  same  plane  with  their  resultant. 

/' 

Of  the   Composition  and  decomposition  of  Forces. 

42.  Not  only  is  it  possible,  by  the  principle  above  established, 
to  reduce  two  concurring  forces  to  one,  and  to  decompose  one  into 
two  others ;  but  we  can  in  general  reduce  to  a  single  force,  as  many 
other  forces  as  we  please,  when  they  are  in  the  same  plane,  or 
when  they  unite  at  the  same  point ;  and  reciprocally,  we  can 
decompose  one  or  several  forces  into  as  many  other  forces  as  we 
please. 

43.  But  before  we   proceed   to  explain  this,  we  m.ust  observe 
Fig.  7.    that  when   a   force  p  acts   upon   a  body   either  by  pushing  or  by 

drawing  it,  it  is  of  no  consequence  at  what  j)oint  of  the  direction  of 
this  force  we  suppose  the  action  to  be  applied.  For  example,  let 
the   force   p  be   exerted    upon   the   body  m  by  means  of   a   rod 


Composition  and  Decomposition  of  Forces.  25 

inflexible  and  without  mass,  or  by  a  thread  inextensible  and 
without  mass,  it  is  the  same  thing,  whether  the  force  p  be  applied 
at  the  point  B,  or  at  the  point  C,  or  whether  it  be  oi'such  a  nature 
as  to  admit  of  being  exerted  at  any  point  D,  on  the  other  side  of 
the  body.  So  long  as  its  action  is  employed  in  the  same  direction, 
the  effect  will  be  the  same.  Distance  can  have  no  influence, 
except  so  far  as  the  action  of  the  power  transmits  itself  by  the  aid 
of  some  instrument,  as  a  lever  or  a  cord,  the  matter  of  which  would 
partake  of  the  action  of  the  power,  all  which  instruments  we  at 
present  leave  out  of  consideration. 

Thus,  if  two  forces  p  and  q,  exerted  in  the  same  plane,  according  Fig.  8. 
to  the  lines  EC,  DB,  draw  or  push  a  body  by  the  two  points  E,  D, 
this  body   is  urged  in  the  same   manner  as  it  would  be,  if  the  two 
forces  were  both  employed  at  their  point  of  meeting  A,  the  direc- 
tions being  supposed  to  remain  unchanged. 

This  being  premised,  we  proceed  to  the  consideration  of  the 
composition  and  decomposition  of  forces. 

44.  Let  there  be  four  forces,  p,  g,  r,  tu,  directed  in  the  manner  Fig.  9. 
represented  in  the  figure,*  and  all  in  the  same  plane.  Let  us  imagine 
the  direction  of  the  force  p,  prolonged  till  it  meets  that  of  the  force 
q  in  the  point  A  ;  AD,  AE,  being  supposed  to  be  the  spaces  that 
the  forces  p,  q,  can  respectively  cause  the  same  body  to  describe,  in 
a  determinate  time,  as  one  second  ;  if  we  form  the  parallelogram 
AEID,  the  diagonal  Al  will  represent  the  resulting  effort  of  p 
and  q,  and  may  consequently  take  the  place  of  these  two  forces. 

Let  us  conceive  now  that  AI  prolonged,  meets  in  B  the  di- 
rection of  the  force  r,  and  having  taken  BL  equal  to  AI,  il  we  take 
BF  for  the  space  that  the  force  r  is  capable  of  making  the  same 
body  describe  in  a  second  ;  the  force  Al  being  supposed  to  be 
applied  at  B,  since  this  force  is  represented  by  BL  =  Ail,  from 
its  action  combined  with  the  force  r  =  BF,  there  will  result  a 
single  force  represented  by  the  diagonal  BG,  of  the  parallelogram 
BLGF.  This  force,  therefore,  will  take  the  place  of  the  forces  r 
and  AI,  that  is,  it  will  take  the  j)lace  of  the  three  forces  r,  q,  and  p. 

Lastly,  let  us  imagine  that  BG  prolonged,  meets  in  C  the 
direction  of  the  force  ct,  and  let  us  make    CK  =  BG.     Let    CH 

*  The  direction   of  a  force  is   indicated  by  the  figure   of  an 
arrow. 
Mech.  4 


26  Statics. 

represent  the  space  which  the  force  zu  is  capable  of  making  the 
same  given  body  describe  in  a  second  ;  then  by  supposing  tlie  force 
BG  =  CK,  applied  at  C  in  the  direction  CG,  from  the  union  of 
this  force  with  the  force  w  there  will  result  a  single  effort  represent- 
ed by  the  diagonal  CJY,  of  the  parallelogrum  CHJVK.  This 
force,  therefore,  will  take  the  place  of  the  forces  n?  and  CK,  or  of 
c7  and  BG  ;  it  will  consequently  take  the  place  of  the  four  forces 
Pf  9^  ^5  ^j  ^""^  '^  accordingly  the  resultant  of  these  four  forces. 

It  is  evident,  therefore,  that  any  number  of  forces,  when  exerted 
in  the  same  plane,  may  be  reduced  to  a  single  force,  and  the 
manner  in  which  this  may  be  done  is  also  manifest. 

45.  It  will  be  seen  moreover  from  the  above  example,  how  we 
may  always  substitute  for  a  single  force  as  many  others  as  we 
please,  and  what  are  the  requisite  conditions  for  effecting  this. 

Instead  of  the  single  force  BG,  for  example,  we  may,  by  form- 
ing the  parallelogram  BLGF,  of  which  BG  \s  the  diagonal,  take 
the  two  forces  represented  by  BF,  BL ;  and  as  we  may  suppose 
each  of  these  two  forces  applied  at  any  such  point  of  their  directions 
respectively  as  we  choose,  we  can  transfer  BL  to  ./?/,  (the  point  A 
being  at  any  assumed  distance  from  B,)  and  form  upon  jil,  another 
parallelogram  AEID  ;  then  for  the  force  AI  may  be  substituted  two 
forces  represented  by  AE,  AD ;  so  that  for  the  single  force 
BG,  we  shall  have  the  three  forces  BF,  AE,  AD,  the  effect  of 
which  will  be  equivalent  to  that  of  BG. 

46.  We  remark  here,  that,  since  there  is  no  other  condition  re- 
quired for  determining  the  forces  AD,  AE,  except  that  they  be 
expressed  by  the  sides  AD,  AE,  of  the  parallelogram  ADIE  of 
which  Al  is  the  diagonal,  which  condition  may  be  fulfilled  in  an  in- 
finite number  of  ways,  whether  the  parallelogram  ADIE  be  in  the 
same  plane  with  the  parallelogram  FBLG  or  in  any  other  plane, 
we  can  decompose  any  force  whatever  BG  into  as  many  others  as 
we  please,  and  which  shall  be  in  such  planes  as  we  please.  We 
shall  see  hereafter  the  use  that  may  be  made  of  this  method  of 
compounding  and  resolving   forces. 

47.  From  what  is  above  said,  it  will  be  perceived  that  we  can 
require  certain  forces  to  pass  through  certain  given  points,  and  even 
to  be  of  certain  determinate  magnitudes,  to  be  parallel  to  certain 
given  lines,  in  a  word,  to  satisfy  certain  given  conditions.     For  ex- 


Composition  and  Decomposition  of  Forces.  27 

ample,  if  we  had  a  force  represented  by  the  line  AB,  and  we  Fi^.  lo. 
would  substitute  two  others,  of  which  one  should  pass  through  the 
point  D,  (in  a  direction  parallel  to  a  line  LX,  whose  position  is 
given,)  and  which  at  the  same  time  should  be  of  a  certain  magni- 
tude LK,  that  is,  such  as  would  cause  a  given  body  to  describe 
LK,  in  the  same  time  in  which  the  force  represented  by  AB  would 
cause  this  same  body  to  describe  the  line  AB  •  the  principles 
above  established  will  enable  us  to  solve  the  problem. 

Through  the  point  D,  we  draw  IC  parallel  to  LX,  and  meeting 
AB  produced  in  some  point  /;  we  take  IC  =  LK,  and  IE  =  AB ; 
then  joining  CE,  we  draw  through  the  point  /the  line  iiiZ  parallel  to 
CE,  and  through  E  the  line  HE  parallel  to  IC;  IC  will  be  the 
force  required,  and  IH  will  be  the  force  which,  combined  with 
IC,  would  take  the  place  of  IE  or  of  AB. 

The  solution  we  have  given  will  always  be  applicable,  except 
when  the  line  LX  is  parallel  to  AB,  and  we  shall  see  soon  what  is 
to  be  done  in  this  case. 

48.    We  remark    further,  that,  the   two  component  forces  p,  q, 
being   represented    by   the   two  sides   AD,  AB,  of  the  parallelo-Fig.5,6. 
gram  DABE,  their  resultant  must  necessarily  be  represented  by  the 
diagonal  AE  of  the  same  parallelogram ;  by  calling  §  the  resultant, 
we  shall  have 

p   :   g   :  :  AD   :   AE, 
q    :  Q    ::   AB    :    AE; 


that  is, 


p    :    q    :    g    ::    AD    :   AB   :   AE, 
::   BE   :   AB   :   AE. 


Now  in  the  triangle  ABE,  we  have 

BE   :   AB    :   AE   ::   sin  BAE   :   sin  BEA   :   sin  ABE.      „ .    „. 

1  rig.  az. 

But  on  account  of  the  parallels  BE,  AD,  the  angle  BEA  =  DAE,^^"""- 
and  ihe  angles  ABE,  BAD,  being  supplements  to  each  other,  Geom. 
sin  ABE  =  sin  BAD;  hence  Tri  .13. 

BE    :   AB    :   AE    ::    sm  BAE    :   sin  DAE   :   shBAD; 
and  consequently 

p    :   q    :   Q    ::   sm  BAE  :   sin   DAE  :  sin  BAD; 
from  which  it  will  be  seen,  that  if  we  suppose  the  force  p  expressed 
by  sin  BAE,  the  force  q  will   be  denoted   by   sin   DAE,   and  the 
force  Q  by  sin  BAD  ;  that  is,  the  two  component  forces  and  the  re- 


28  Statics. 

sultant   may  be  represented  each  by  the  sine  of  the  angle  compre- 
hended between  the  directions  of  the  two  others. 

In  representing  forces,  therefore,  we  may  employ  indifferently 
either  the  lines  taken  in  the  directions  of  these  forces,  or  the  sines 
of  the  angles  comprehended  between  these  directions,  provided  we 
take  for  each  the  sine  of  the  angle  comprehended  between  the  di- 
rections of  the  two  others. 

Tliis  last  method  of  expressing  forces  has  its  peculiar  advan- 
tages, as  we  shall  see  in  what  follows. 

Fig.  11,  49.  If  from  the  point  ./3  as  a  centre,  and  with  any  radius  AC, 
we  describe  an  arc  of  a  circle  HCG,  meeting  in  G  and  H  the  di- 
rections of  the  forces^,  q,  and  let  fall  from  the  point  C  upon  AD, 
AB,  the  perpendiculars  CF,  CI,  and  from  the  point  H  upon  AD 
the  perpendicular  HL,  it  will  be  readily  seen  that  CF,  CI,  HL, 
are  the  sines  of  the  angles  DAE,  BAE,  BAD,  respectively  ;  we 
have  accordingly 

p  :  q  :  Q  :  :    CI  :   CF  :  HL. 

Fig.  13,  50.  Let  us  suppose  now,  that  while  the  directions  of  the  two 
forces  p,  q  pass  through  the  two  fixed  points  K,  JV,  their  point  of 
meeting  *^  is  removed  further  and  further;  it  is  evident  that  the 
sines  of  the  angles  BAE,  DAE,  BAD,  will  approach  more  and 
more  to  a  coincidence  with  the  arcs  CH,  CG,  HG  ;  if  therefore 
the  point  A  is  removed  to  an  infinite  distance  from  the  fixed 
points  K,  JV,  CF,  CI,  HL,  will  coincide  with  the  arc  HG, 
which  in  this  case  becomes  a  straight  line  perpendicular  to  the 
two  lines  AK,  AJV,  which  are  then   parallel  to  each  other  and  to 

Geom.     the  line  AE ;  and,  since  we  have  always 

^^  p  :  q  :  q  '.  :    CI  :   CF  :  HL, 

HL=CI-\-  CF  {fig.  13). 
HL  =  CI—  CF  (fig.  14). 

Fig.  15,  We  conclude,  therefore,  that  when  two  forces  p,  q,  are  exerted  in 
parallel  directions, 

1 .  That  their  resultant  is  in  a  direction  constituting  another 
parallel ; 

2.  That  if  we  draw  a  line  Fl  perpendicular  to  these  directions, 
each  of  the  forces  will  be  represented  by  the  part  of  this  perpendicu- 
lar comprehended  between  the  directions  of  the  tivo  others  ; 


Composition  and  Decomposition  of  Forces.  29 

3.  That  the  resultant  is  equal  to  the  sum  of  the  two  components, 
when  these  act  the  same  way,  and  to  their  difference,  when  their  ac- 
tion is  opposed  the  one  to  the  other. 

51.  Since  we  have 

p    :    q    :    g    ::    EI  :    EF  :   FI, 

we  have 

p  :  q  :  :  El  :  EF,     and    p  :  q  :  :  EI  :  Fl; 

that  is,  of  two  parallel  component  forces  and  their  resultant,  either 
two  are  to  each  other  reciprocally,  as  the  two  perpendiculars  let 
fall  upon  their  directions  respectively  from  the  same  point  in  the 
direction  of  the  third. 

52.  If  we  draw  arbitrarily  any  line  AB  C,  we  shall  have  jgg. 

BC  :  JlB  :  AC  ::  EI  :  EF  :  FI, 

and  consequently 

p  :  q  :  g  ::  BC  :  AB  :  AC; 

that  is,  if  a  straight  line  be  drawn  at  pleasure,  cutting  the  direc- 
tions of  two  parallel  forces  and  their  resultant,  each  of  these  forces 
will  be  represented  by  that  part  of  the  straight  line  which  is  compre- 
hended between  the  directions  of  the  two  others. 

53.  It  will  hence  be  readily  perceived  how  we  ought  to  pro- 
ceed in  order  to  find  the  resultant  of  several  parallel  forces ;  and 
reciprocally,  how  we  can  substitute  for  a  single  force,  any  number 
whatever  of  parallel  forces. 

If,  for  example,  it  were  proposed  to  reduce  to  a  single  force 
the  two  parallel  forces  p,  q,  which   act  the  same  way;    any  straight  pig.  15. 
line  ^BC  being  drawn;  as  the  resultant  q  is   equal  to  p  -j- 9'»  it  is      50 
only  necessary   to  find   the  point  B  through  which  this  resultant 
must  pass.     Now  we  have  50. 

2?    :        g        :  :     BC    :    AC, 
that  is, 

p    :   p   +  q    ::    BC    :    AC. 
We  have  therefore  only  to  take  between  the  two  points  A,  C,  a 

point  B  such  that  BC  shall  be  equal  to  ?-^^-^.  Geom. 

If  the  two  parallel  forces  are  opposed  to  each  other,  the  resultant  Fig.  16. 
will  be   equal   to   their  difference  p  —  q,  or  q  —  p.     Suppose  p      50. 
greater  than  q.     Having  drawn  the  line  AC  at  pleasure,  it  will  be 


30  Statics. 

necessary  to  prolong  AC  beyond  A,  with  respect  to  C,  by  a  quan- 
tity AB,  such  that  we  shall  have 
52.  p  :       Q        ::  BC  :  AC 

or 

p   :  p  —  q  ::  BC  :  AC; 

in  other  words,  it  is  necessary  to  take  BC  equal  to^- . 

If  q  is  greater  than  p,  the  point   B  will  be   in  AC  produced  be- 
yond C  with  respect  to  A. 

Fig.  17.  54.  If  we  had  a  third  force  r,  we  should  first  find  the  resultant  ^' 
of  the  two  forces  p,  q,  and  then  seek  the  resultant  q  of  the  two  forces 
^'  and  r,  as  if  there  were  only  these  two  ;  that  is,  we  should  proceed 
in  precisely  the  same  manner  as  we  have  done  in  the  preceding 
article. 

Fig.  15  55.  Hence,  reciprocally,  if  we  would  decompose  any  force  g 
^^'  into  two  others  parallel  to  it,  we  should  take  arbitrarily  a  line  AF 
parallel  to  the  direction  of  ^  ;  and  having  assumed  this  line  as  the 
direction  of  one  of  the  components,  we  take  arbitrarily  for  the  value 
of  this  component  any  quantity  p  smaller  than  q,  if  it  is  proposed 
that  the  two  components  should  act  on  opposite  sides  of  the  force  g  ; 
the  second  component  q  must  in  this  case  be  equal  to  g  —  p;  and 
in  order  to  find  its  position,  it  is  only  necessary,  having  drawn  any 
straight  line  CBA,  to  take  in  AB  produced  the  part  £C,  such  as  to 
give  the  proportion 

q  :p  I'.AB  :BC; 
then  through  the  point  C  we  draw  IC,  parallel  to  EB,  and  this  will 
be  the  direction  of  the  foi»ce  q. 

But  if  the  two  component  forces  are  required  to  be  on  the  same 
side  (in  which  case  they  will  be  directed  opposite  ways),  then  we 
take  for  p  any  quantity,  whether  greater  or  less  than  g,  and  if  it  he 
greater  it  will  be  directed  the  same  way  with  g,  and  if  less,  it  will 
have  a  contrary  direction  with  respect  to  g.  Having  drawn  a  line 
AF  parallel  to  EB,  as  the  direction  of  p,  we  take  upon  any  as- 
sumed line  BA  C  the  point  C,  such  as  will  give 

p  —  g  or  g — p   :  g  ::  AB  :  AC ', 
and  C  will  be  the  point  through  which  the  force  q  must  pass  par- 
allel to  the  given  force  g  ;  and  the  point  C  will  be  beyond  A  with 


Composition  and  Decomposition  of  Forces.  31 

respect  to  B,  when  p  is  greater  than  q  ;  and  it  will  be  between  A 
and  B  when  p  is  less  than  g. 

56.  Since  what  we  have  now  said  of  the  force  g  with  respect 
to  the  components  p,  q,  may  evidently  be  applied  to  each  of  these 
latter  forces,  it  will  be  seen  how  we  may  substitute  for  any  single 
force,  as  many  others  as  we  please,  the  directions  of  which  are 
parallel. 

Of  Moments  and  their  Use  in  the  Composition  and  Decomposition 

of  Forces. 

57.  The  propositions  we  have  established,  are  sufficient  for  the 
composition  and  decomposition  of  forces,  whatever  be  their  magni- 
tudes and  directions,  provided  they  act  in  the  san)e  plane.  But  the 
different  kinds  of  motion  which  we  have  to  consider,  require  more 
simple  and  more  expeditious  means  for  determining  the  resultant 
of  forces,  and  its  direction. 

58.  If  from  any  point  F  taken  in   the  plane  of  any  paraUelo-  Fig.  18, 
gram  ABCD,  we  let  fall  upon  the  contiguous  sides  AB,  AD,  and 

the  diagonal  AC,  the  perpendiculars  FE,  FH,  FG,  the  sum  of  the 
products  of  the  two  contiguous  sides  by  the  perpendiculars  respective- 
ly let  fall  upon  them,  ivill  be  equal  to  the  product  of  the  diagonal  by 
its  perpendicular,  when  the  point  F  is  neither  in  the  angle  BAD,  Fig.  18. 
nor  in  the  vertical  angle  KAL.  If,  on  the  contrary,  the  point  F  is  p-ig.  19. 
either  in  the  angle  BAD,  or  in  the  vertical  angle  KAL,  the  differ- 
ence of  the  products  of  the  two  contiguous  sides  by  their  respective 
perpendiculars  ivill  be  equal  to  the  product  of  the  diagonal  by  the 
perpendicular  let  fall  upon  it. 

Produce  the  side  BC  till  it  meets  in  I  the  perpendicular  FH, 
and  join  FA,  FB,  FC,  FD.     The  triangle 

FAC=FAB  +  ABC  +  FBC  =  FAB  +  ADC  +  FBC.   rig.  is. 
Now 

1 .  The  triangle  FAC  =  d£xF^^  ^^^^ 

176. 

2.  The  triangle  FAB  =  ^^  ^  ^^. 

3.  The  triangle  ADC  having  AD  for  its  base,  and  III  for  its 
altitude, 

4t 


32  Statics. 

4.   Ihe  triangle  FBC  =. ^ =  — . 

Whence 

AC  X  FG  __AB  X  FE       AD  x  IH   ,    AD  x  FI 
2  -          ~2  +  2  "i  2       • 

Now  IH  +  FI  =  FH;     therefore,  hy  doubling  the  whole,    we 
have 

AC  X  FG  =  AB  X  FE  +  AD  X  FH. 
Fig.  19.        With  respect  to  the  triangle  FAC,  we  have 

FACz=ABC  —  FAB  — FBC  =  ADC  — FAB— FBC, 
that  is, 

AC  X  FG  _AD  X  IH       AB  x  FE       BC  X  FI ^ 
2  ~  2  2  2  ' 

or,  since  BC  =  AD,  and  IH —  J7  =  FH,  the  whole  being  dou- 
bled, 

AC  X  FG  =  AD  X  FH  —  ABx  FE. 

59.  Since  we  have  before  shown  that  any  two  forces  and  their 
resultant  may  be  represented  by  the  sides  and  diagonal  of  a  par- 
allelogram, formed  upon  the  directions  of  these  forces,  if  p,  q,  be 
two  forces,  represented  by  the  lines  AB,  AD,  in  which  case  their 
resultant  q  would  be  represented  by  AC,  any  point  F  being  taken 
in  the  plane  of  these  three  forces  without  the  angle  BAD,  and 
without  the  vertical  angle  KAL,  we  have  always 

gXFG  =  p  XFE+  qx  FH', 
and  when  the  point  F  is  taken  in  the  angle  BAD,  or  in  the  vertical 
angle  KAL,  we  shall  liave  in  like  manner 

qX  FG  =  qx  FH—p  xFE. 

60.  The  product  of  a  force  by  the  distance  of  its  direction  from 
a  fixed  point  is  called  the  moment  of  this  force.  Thus  q  X  FH  is 
the  moment  of  the  force  q ;  and  q  X  FG  is  the  moment  of  the 
force  Q. 

61.  Asa  force  is  estimated  by  its  quantity  of  motion,  that  is, 
by  the  product  of  a  determinate  mass  into  the  velocity  which  it  is 
capable  of  giving  to  this  mass,  the  moment  of  any  force  has  for  its 
measure  the  product  of  a  mass,  by  its  velocity,  and  by  the  distance 
of  its  direction  from  a  fixed  point. 

62.  If  the  perpendiculars  FH,  FG,  FE,  are  considered  as 
lines  inflexible  and  without  mass,  connected  together  and  fixed  to 


Composition  and  Decomposition  of  Forces.  33 

the  point  F,  in  such  a  manner  as  to   admit  only  of  their  turning 

about  this  point ;  and  we  suppose   that  the  forces  p,  q,   and  their 

resultant  g,  are  applied  at  the  extremities  E,  H,  G,   we  shall  see 

that  these  three  forces    tend  each  to  turn  the   system   in  the  same„ 

Fig.  18. 
direction  about  the  point  F :  and  that  the  two  forces   q,  p,  tend  to 

.  .  Fiir.  19. 

turn   the    system  in   a  different   direction    from  that  in   which  the    °' 

force  p  tends  to  turn  it. 

We  infer,  therefore,  that  the  moment  of  the  resultant,  taken  with 
respect  to  any  fixed  point  F,  is  .always  equal  to  the  sum  or  to  the 
difference  of  the  moments  of  the  two  components,  according  as  these 
components  tend  to  turn  the  body  or  system  in  the  same  direction,  or 
in  opposite  directions,  about  this  fixed  point. 

Go.  We  conclude,  moreover,  that  in  general,  whatever  be  the 
number  of  forces  p,  q,  r,  sr,  ^-c,  and  whatever  their  magnitudes  and  Fig.  Q 
directions,  provided  they  act  in  the  same  plane,  the  moment  of  the 
resultant  of  all  these  forces,  t.iken  with  reapect  to  a  fixed  point  F, 
assumed  at  pleasure  in  this  plane,  will  always  be  equal  to  the  sum  of 
the  moments  of  the  forces  tending  to  turn  the  system  in  one  direction 
about  this  point,  minus  the  sum  of  the  moments  of  those  which  tend 
to  turn  it  in  the  opposite  direction. 

Indeed,  if  we  suppose  that  g  is  the  resultant  of  the  two  forces  Fig.  20. 
p,  q,   g"  that  of  g    and  r,  and  g   that  of  g"  and  nr ;  if  we  suppose, 
moreover,  that  ^  represents  the  moment  of  g,  and  ^i  tbat  of  g",  then 
by  letting  fall  the  perpendiculars  F.4,  FE,  FG,  FD,  FB,  upon  the 
components  p,  q,  r,  vj,  and  their  resultant  g,  we  shall  have 

1.  ^  =  p  X  FA  +  qX  FE, 

2.  ^i'  =  fi  —  r  X  FG, 

3.  g  X  FB  =  ^'  —  r^  X  FD; 

adding  therefore  these  three  equations  together,  and  suppressing 
those  quantities  that  cancel  each  other  in  the  two  members,  we  shall 
have 

g  X  FB  =  p  X  FA  -\-q  X  FE  —  r  X  FG—  ^  X  FD ; 
from  which  it  will  be   seen,   that  the   moments  of  the  two  forces 
r,  c7,  which  tend  to  turn  the  system  from  right  to  left,  are  of  a  con- 
trary sign  to  that  of  the  forces  p,  q,  which  tend  to  turn   it  from  left 
to  right. 

64.  If  the  point  F  were  exactly  in  the  direction  of  the    resul- 
tant, the  moment  of  this  force  would  be  zero  ;  but,  since  it  is  equal 
Mech.  5 


34  Statics. 

to  the  sum  of  the  moments  of  the  forces  which  tend  to  turn  the 
system  in  one  direction,  minus  the  sum  of  the  forces  tending  to 
turn  it  in  the  opposite  direction,  we  conclude  that  the  difference  of 
these  two  sums  of  moments,  taken  with  respect  to  any  point  what- 
ever in  the  direction  of  the  resultant,  is  zero. 

And  reciprocally,  if  the  sum  of  the  moments  of  the  several  forces 
which  tend  to  turn  a  system  about  a  given  point,  minus  the  sum  of 
the  moments  of  those  ichich  tend  to  turn  it  in  the  opposite  direction 
about  this  same  point,  is  zero  ;  it  must  be  inferred,  that  the  resultant 
passes  through  this  point. 

65.  As  these  propositions  hold  true,  whatever  be  the  angles 
formed  by  the  directions  of  the  forces,  they  are  applicable,  when 
these  angles  are  infinitely  small,  that  is,  when  the  directions  of  the 
forces  are  parallel. 

66.  We  may  thus  derive  a  very  simple  method  of  obtaining  the 
position  and  magnitude  of  the  resultant  of  any  number  of  forces, 
when  they  all  act  in  the  same  plane. 

Let  us  suppose,  in  the  first  place,  that  they  are  all  parallel ;  and, 
not  to  make  the  pioblem  more  com|jlicated  than  is  necessary,  let 
us  suppose  that  theie  are  only  three  forces  ;  it  will  be  easily  inferred 
how  we  are  to  proceed  in  case  of  a  greater  number. 

Fig.  21.  Accordingly,  let  there  be  the  three  known  forces  p,  q,  r,  the 
two  first  being  directed  the  same  way,  and  the  third  having  a  con- 
trary direction.  Having  drawn  arbitrarily  any  line  FABC,  perpen- 
dicularly to  the  directions  A  p,  B  q,  Stc,  we  will  suppose  that  D  is 
the  point  through  which  the  resultant  g  is  to  pass.  Then,  having 
taken  at  pleasure  a  point  J^  in  FABC,  we  shall  have,  according  to 
63.     what  has  been  demonstrated, 

p  X  FAi-q  X  FB  —  r  X  FC  =  q  X  FD. 

Now  the  distances  FA,  FB,  FC,  and  the  forces  p,  q,  r,  being 
known,  it  will  be  easy  to  deduce  from  the  above  equation,  the  value 
of  the  distance  FD,  through  which  the  resultant  would  pass,  if  the 
value  of  this  resultant  g  were  known.  In  order  to  find  it,  we  take 
another  point  go  in  AF  produced,  and  by  proceeding  as  above,  we 
have 

p  X  ^A  -\-  q  X  (pB  —  r  X  <fCz=q  X  (pD. 


Composition  and  Decomposition  of  Forces.  35 

If  from  this  second  equ^^lion  we  subtract  the  first,   recollecting  that 
<pA  —  FA  =  cpF,     cpB  —  FB  =  cfF, 
cpC  —  FC  =  (pF,    <pD  —  FD  =  <fF, 

we  shall  have 

p  X  (fF  -\-  q  X  (fF—r  X  (pF  =  Q  X  (pF  ; 
that  is,  the  whole  being  (llvideci  by  cpF,  • 

2^  +  9  —  >'  =  Q- 
If  we  examine  the  process  now  pursued,  we  shall  see  that  it 
does  not  depend  in  any  degree  upon  the  number  of  forces,  but  that 
it  is  applicable,  whatever  this  number  may  be.  We  must  infer 
therefore,  that  the  resultant  of  any  number  of  parallel  forces  is 
equal  to  the  sum  of  those  which  act  in  one  direction  minus  the  sura 
of  those  which  act  in  the  opposite  direction. 
If  now  in  ihe  equation 

p  X  FA  +  q  X  FB  —  r  X  FC=o  X  FD, 
found  above,  we  put  for  §  its  value  p  -{-  q  —  r,  just  obtained,   we 
shall  have 

p  X  FJ  Jr  q  X  FB  —  r  X  FC  =  (p  -{-  q  —  r)  X  FD, 
from  which  we  deduce 

p  X  FA  -{-qXFB  —  r  X  FC 
ji_U  =.  , — - — ; 

p  +  q  —  r 

or,  bearing  in  mind,  that  the  process  by  which  we  have  arrived  at 
this  result,  does  not  depend  upon  the  number  of  forces  employed, 
we  infer,  as  a  general  conclusion,  that  in  order  to  determine  at  what 
distance  from  a  given  point  the  resultant  of  several  parallel  forces 
passes,  from  the  sum  of  the  moments  of  the  forces  which  tend  to 
turn  the  system  in  one  direction,  we  must  subtract  the  sum  of  the  mo- 
ments of  the  forces  tending  to  turn  it  in  the  opposite  direction,  and 
divide  the  remainder  by  the  sum  of  the  forces  which  act  in  one  direc- 
tion, minus  the  sum  of  those  which  act  in  a  contrary  direction.* 

*  We  must  take  care  not  to  confound  the  forces  which  act  in 
opposite  directions,  with  those  which  tend  to  turn  the  system  in 
opposite  directions.  Two  forces  which  act  in  opposite  directions 
often  tend  to  turn  the  system  in  the  same  direction.  This  depends 
upon  the  point  to  which  the  rotation,  or  the  moments,  is  referred. 
The  two  forces  q,  r,  for  example,  act  in  opposite  directions,  but  Fig.  21. 
they  both  tend  to  turn  the  line  JBC  in  the  same  direction,   about   a 


3G  Statics. 

G7.  If  die  point  F,  assumed  arbitrarily,  should  happen  to  be  so 
taken  as  to  fall  in  D,  through  which  the  resultant  passes,  the  dis- 
tance FD  being  zero,  its  value 

p  X  FA  -}-  g  X  FB  —r  X  FC 
ji  -\-  g  —  ^  ' 

since  the  force  p  tends  to'turn  the  system  about  die  point  D  in  a 
direction  opposite  to  that  in  which  the  force  q  tends  to  turn  it, 
becomes 

—p  X  DA-^q  X  DB—r  X  DC 
p  -t-  q  —  r  ' 

and  is  equal  to  zero  ;  we  have  consequendy 

—p  X  DA-\-  qX  DB—r  x  DC  =  o, 

or 

q  X  DB  =  p  X  DA  +  r  X  DC. 

44.  Moreover,  as  die  point  F,  taken  arbitrarily,  may  be  higher  or  lower, 
as  we  please,  the  point  D  has  not  been  supposed  to  be  in  one  point 
of  the  direcdon  of  the  resultant  rather  than  in  another ;  it  follows, 
therefore,  that  the  moments  of  several  parallel  forces,  taken  with 
respect  to  any  point  whatever  in  the  direction  of  the  resultant,  are 
such,  that  the  sum  of  the  moments  of  the  forces  which  tend  to  turn 
the  system  in  one  direction,  is  always  equal  to  the  sum  of  the  mo- 
ments of  those  which  tend  to  turn  it  in  the  opposite  direction. 

68.  Therefore  by  taking  with  contrary  signs  the  moments  of  the 
forces  which  tend  to  turn  the  system  in  opposite  direcdons,  and  by 
taking  also  with  contrary  signs  the  forces  which  act  in  opposite 
directions,  we  may  infer  as  a  general  conclusion  ; 

1.  That  the  resultant  of  any  number  whatever  of  parallel  forces 
is  always  equal  to  the  sum  of  all  these  forces  ; 

2.  That  this  resultant,  which  is  parallel  to  the  component  forces, 
passes  through  a  series  of  points  each  of  which  has  this  property,  that 
the  sum  of  the  moments,  taken  ivith  respect  to  this  point,  is  zero. 

The  above  proposidons  are  of  the  greatest  importance.  We 
shall  see  soon  with  wdiat  facility  they  enable  us  to  find   the  centre 

point  taken  between  B  and  C ;  and  if  we  consider  the  rotation 
with  reference  to  the  point  F,  the  force  q  tends  to  turn  FC  in  a 
direction  opposite  to  that  in  which  the  force  r  tends  to  turn  it. 


Composition  and  Decomposition  of  Forces,  37 

of  gravity  of  bodies.      We  proceed  now  to  the  consideration  of 
forces  the  directions  of  which  are  inclined  to  each  other. 

69.  Let  there  be  any  number  of  forces  p,  q,  r,  &ic.,  all  exert- ^'S- 22- 
ed  in  the  same  plane,  let  the  force  p,  acting  according  to  AE,  be 
represented  by  AE,  and  let  the  force  q,  acting  according  to  BG, 
be  represented  by  BG,  and  the  force  r,  acting  according  to  CL,  be 
represented  by  CL.  Through  a  point  F,  laken  arbitrarily  in  the 
plane  of  these  forces,  suppose  two  straight  lines  FO,  FB" ,  mak- 
ing any  angle  with  each  other  (and  for  the  sake  of  greater  sim- 
plicity, let  this  angle  be  a  right  angle)  ;  and  let  us  imagine  the 
forces  p,  q,  r,  or  AE,  BG,  CL,  decomposed  each  into  two  others, 
one  of  which  shall  be  parallel  to  FO,  and  the  other  to  FB"^  and 
which  will  consequently  be  represented,  each  by  the  correspond- 
ing side  of  the  parallelogram,  the  diagonal  of  which  represents  the  ^0. 
given  force. 

It  is  clear  from   what  has  been  said,  that  the  forces  AV,  BH,  66,68. 
CK,  being  parallel,  will  have  for  their  resultant  a  single  force  DN, 
parallel  to  AV,  BH,  &c.,  the  value  of  which  will  be 
.     AV+  BH-\-  CK,"" 

and  which   will  pass  at  a  distance  D'D  from  JPC,  equal  to  the  ex- 
pression below,  namely, 

D'D  —  ^^X  AA'  +  BH  X  BB'  -\-  CK  x  CC 
~  AV-i-  BH  +  CK 

In  like  manner,  the  forces  AT,  BR,  CM,  parallel  to  FB",  are 
reduced  to  a  single  one  DO,  parallel  to  AI,  he,  and  equal  to 
AI  -\-  BR  —  C31,  and  which  (by  supposing  that  D  is  the  point 
where  the  direction  of  this  force  meets  that  of  the  force  ND)  will 
pass  at  a  distance  D"D  from  FB'',  equal  to  the  following  expres- 
sion, namely^ 

jy„jy  _  AT  X  AA"  +  BR  X  BE"  —  C3I  x  CC 
AI  -\r  BR  —  CM 

*  We  must  not  lose  sight  of  what  was  said  art.  39.  By  the 
forces  AE,  BG,  doc,  we  are  to  understand  that  the  lines  AE, 
BG,  di-c,  are  to  each  other  as  the  quantities  of  motion  capable  of 
being  produced  by  the  forces  p,  q,  &c.,  in  the  masses  to  which 
they  are  applied.  It  is  to  be  observed,  likewise,  with  respect  to  the 
forces  A  V,  BH,  &.c.,  tliat  we  mean  by  them  quantities  of  motion, 
which  are  to  the  quantities  of  motion  represented  by  AE,BG,6lc., 
tis  AV,  BH,  &LC.,  are  to  AE,  BG,  &,c.,  respectively. 


38  Statics. 

This  being  supposed,  the  forces  p,  q,  r,  and  their  directions, 
(that  is,  the  angles  which  they  make  witli  the  known  fixed  lines 
FO,  FB",  or  with  their  parallels,)  being  considered  as  known,  we 
know  in  each  of  the  triangles  AEI,  BGR,  CLK,  the  hypothenuse 
and  the  angles.  It  will  accordingly  be  easy  to  determine  the  lines 
Trig.  30.^/,  BR,  KL,  or  CM,  and  the  lines  IE  or  AV,  RG  or  BH, 
and  CK.  We  shall  consequently  know  the  values  of  the  two  re- 
sultants AV-^  BH  +  CK,  and  AI  +  BR  —  CM.  Moreover, 
as  we  cannot  but  know  the  distances  AA',  AA",  BB',  BB",  &-c., 
since  the  position  of  the  points  A,  B,  he,  where  the  forces  are 
applied  are  supposed  to  be  given,  we  are  acquainted  with  all  the 
quantities  which  enter  into  the  expression  of  the  distances  D'D, 
D'^D.  It  will  be  easy,  therefore,  to  determine  the  point  D,  where 
these  two  resultants  meet.     Accordingly,  taking 

DO  =  AI+BR—CM,  and  DN=AV+  BH-\-  CK, 
and  forming  the  parallelogram  DNTO,  we  shall  have  the  diago- 
nal  DT  for  the   resultant  q,  of  the  two  partial  resultants,  parallel 
to   FC  and   FB",  that  is,  for  the   resultant  of  all  the  proposed 
forces.  -^ 

Of  Parallel  Forces  which  act  in  different  Planes. 

Fig.  23.  70.  Let  there  be  three  forces  p,  q,  r,  directed  according  to  the 
lines  Ap,  B  q,  C  r,  parallel  to  each  other,  but  situated  in  different 
planes. 

Imagine  a  plane  JLZ  to  which  the  three  straight  lines  A  p,  Sic, 
are  perpendicular,  and  another  plane  ZV  to  which  they  are  par- 
allel, and  let  A,  B,  C,  be  the  three  points  where  these  lines  meet 
the  plane  XZ. 

Geom.  '^'i^  ^^'°  forces  p,  r,  are  in  the  same  plane,  the  intersection  of 

335,324.  vvhich  with  the   plane   XZ  is  the  straight  line  AC.     These  two 

50.     forces  may  therefore  be  reduced  to  a  single  one  §',  equal  to  ^  -j-  r, 

67.     having  a  direction  parallel  to  that  of  the  components,  and  passing 

through  a  point  D,  such  that  p  X  AD  =  r  X   CD. 

The  two  forces  q',  q,  are  in  the  same  plane,  the  intersection 
of  which  with  the  plane  XZ  is  BD.  These  may  accordingly  be 
reduced  to  a  single  one  q,  equal  to  /  -f"  5'j  that  is,  equal  to 

F  +  q  +  r, 
having  a  direction  parallel  to  that  of  §'  and  r,  and  passing  through 


Parallel  Forces  acting  in  different  Planes.  39 

a  point  E,  such  that  /  X  BE  —  q  X  BE.  It  follows,  therefore, 
from  this  and  what  is  said  above,  that  any  number  of  forces,  the 
directions  of  which  are  parnllel,  jnay  be  reduced  to  a  single  one, 
equal  to  the  sum  of  those  which  act  in  one  direction,  minus  the  sum 
of  those  which  act  in  the  contrary  direction,  whether  ihe  given  forces 
are  in  the  same  or  in  different  jjlanes. 

Let  us  now  inquire  more  particularly  how  we  are  to  determine 
through  what  point  the  resultant  passes. 

If  from  the  points  A,  D,  C,  B,  E,  we  draw  the  lines  AA', 
DD',  CO,  BB',  EE',  perpendicularly  to  the  common  intersection 
of  the  two  planes  ^Z,  ZV;  on  account  of  the  parallels  AA'.  DD', 
CO,  we  shall  have 

AD:  CD  ::  AD'  :  OD', 
Now  the  equation  p  X  AD  =  r  X   CD,  found  above,  gives 

AD:CD::r:j); 
hence 

A'D'  :  OD'  ::  r  :  p, 
and  consequently 

p  X  A'D'  =zr  X   CD'. 

In  like  manner  from  the  parallels  DD',  EE,  BB',  we  ob- 
tain, 

DE  :  BE  ::  D'E  :  BE ; 

and  from  the  equation  ^'  X  DE  =  q  X  BE,  we  have  the  pro- 
portion 

DE  :  BE  ::  q  :  q'; 

therefore 


whence 


DE  :  BE'  '.'.  q  :      ^/ 

::  q  :  p  -\-r', 


(p  -f  r)  X  D'E  =  q  X  BE'. 

Let  us  now  take  in  the  intersection  ZF  oi  the  two  planes,  a 
fixed  point  F,  and  seek  the  distance  FE'  of  this  point  from  the 
point  E,  corresponding  to  E,  through  which  the  resultant  passes. 
It  is  clear  that 

A'D'  =  FD'  —  FA',      CD'  =  FO  —  FD', 
D'E  =  FE  —  FD',     BE  =  FB'  —  FE. 

Substituting  these  values  for  their  equals  in  the  two  equations, 


40  Statics. 

p  X  A'D'  =  r  X    CD',     (p  +  r)  X  BE'  =  q  X  B'B, 
we  shall  have 

p  X  FD'  —px  FA'  =  r  X  FC'  —  r  X  FD', 

and 

{p^r)  X  FE'—{p  +  r)  FD'  =  q  X  FB' —  q  x  FE'. 

The  first  of  these  two  equations  gives 

{p  +  r)  X  FD'  =  p  X  FA'  +  r  X  FC ; 
substituting    this   value    for  its  equal    in  the    second    equation,  we 
obtain 

{p-\-r)  X  FE'—pxFA'  —  rX  FC  =  q  X  FB'  —  qxFE'-, 

or,  the  terms  multiplied  by  FE'  being  collected  into  one  factor, 

(P  +  ?  +  0  X  FE'  =px  FA'  -\-  q  X  FB'  +  rx  FC, 

which  gives 

p^,  _  pX  FA'  +  qX  FB'  +  rX  FC 
p-\-q-\-r 

Now  this  expression  for  the  distance  FE'  is  precisely  that  which 
we  should  have  found  for  the  distance  at  which  the  resiiltant  passes, 
if  the  three  forces  p,  q,  r,  had  all  been  in  the  plane  ZV,  and  had 
passed  through  the  points  A',  C,  B',  corresponding  to  the  points 
A,  C,  B,  through  which  they  actually  piiss.  If,  therefore,  we 
imagine  the  straight  line  FX  perpendicular  to  the  plane  ZV,  we 
shall  find  the  distance  FE,  of  the  resultant  from  this  straight  line, 
by  taking  the  sum  of  the  moments  with  reference  to  this  line  (as  if 
the  forces,  retaining  their  distances  respectively  from  this  line,  were 
all  in  the  plane  ZF,  to  which  this  line  is  perpendicular),  and  di- 
viding this  sum  of  the  moments*  by  the  sum  of  the  forces. 

To  determine  the  point  E,  therefore,  it  only  remains  to  find  the 
distance  EE,  or  (by  taking  E^''  parallel  to  ZF)  the  distance  FE", 
at  which  this  same  force  passes,  from  ZF.  Now  it  is  manifest 
from  what  we  have  said  with  respect  to  the  distance  FE',  that  in 

*  It  must  be  observed,  once  for  all,  that  by  the  general  term, 
sum  of  the  moments,  is  to  be  understood  the  sum  of  the  moments 
of  the  forces  that  tend  to  turn  the  system  in  one  direction,  minus 
the  sum  of  those  which  tend  to  turn  it  in  the  opposite  direction. 
By  sum  of  the  forces,  also,  is  to  be  understood  the  sum  of  those 
which  act  in  one  direction,  minus  the  sum  of  those  which  act  in 
the  opposite  direction. 


Forces  not  parallel  acting  in  different  Planes.  41 

order  to  find  the  distance  FE^',  we  have  only  to  imagine  a  plane 
passing  through  ZF,  perpendicular  to  the  direction  of  the  forces, 
and  then  to  take  the  sum  of  the  moments  with  respect  to  ZF  (the 
intersection  of  this  plane  with  the  plane  ZV),  as  if  the  forces, 
without  changing  their  distances  respectively  from  the  plane  ZV, 
were  all  in  the  plane  XV,  and  to  divide  this  sum  of  the  moments 
by  the  sum  of  the  forces.  We  should  then  have  every  thing  which 
is  necessary  for  fixing  the  point  E,  through  which  the  resultant 
passes.  Top. 


Of  Forces  the  Directions  of  which  are  neither  in  the  same  Plane  nor 
parallel  to  each  other. 

71.  Let  p,  g,  r,  be  three  forces  directed  in  the  manner  repre- ^'S- 24. 
sented  in  the  figure,  and  situated  in  three  different  planes.  Sup- 
pose any  plane  XZ  meeting  in  H  tiie  direction  of  jj,  in  F  the  di- 
rection of  q,  and  in  L  the  direction  of  r.  As  a  force  may  be  con- 
sidered as  applied  at  any  point  whatever  in  its  direction,  let  us 
suppose  the  three  forces  to  be  applied  at  the  points  H,  F,  L,  and 
to  be  represented  by  the  lines  HV,  FT,  LK,  prolongations  respec- 
tively of  the  directions  of  the  several  forces  below  the  plane  XZ. 
Let  us  also  imagine  planes  passing  through  the  lines  AH,  BF, 
CL,  perpendicularly  to  the  plane  XZ,  the  intersections  with  XZ 
being  represented  by  the  straight  lines  GHN,  EFY,  DLM.  This 
premised,  it  is  evident  that  we  may  decompose  each  of  the  forces 
in  question  into  two  others,  one  of  which  shall  be  in  the  plane  XZ, 
and  the  other  perpendicular  to  this  plane.  We  can,  for  example, 
decompose  the  force  HV  into  two  others,  one  directed  according 
to  HN,  and  the  other  according  to  HO,  perpendicular  to  the  plane 
XZ  ;  so  that  for  the  three  forces  HV,  FT,  LK,  may  be  substituted 
the  six  forces 

HN,  FY,  LM,  HO,  FS,  LI, 
the  three  first  of  which  are  in  the  plane   XZ,  and  the   three  last 
perpendicular  to  this  plane. 

Now  the  three  forces  HN,  FY,  LM,  may  be  reduced  to  a 
single  one,  which  shall  also  be  in  the  plane  XZ;  and  the  three  HO,    69. 
FS,  LI,  may  likewise  be  reduced  to  a  single  one,  which  shall  be 
perpendicular  to  the  plane  XZ.     Accordingly,  whatever  be  the  num-    70. 
ber  of  given  forces,  and  whatever  their  directions,  we  may  always 
Mech.  6 


42  Statics. 

reduce  them  to  two  at  the  most,  one  being  in  the  direction  of  a  known 
plane,  and  the  other  perpendicular  to  this  plane. 

Although  the  demonstration  of  this  proposition  may  appear  to 
be  adapted  only  to  those  cases  where  all  the  forces  meet  the  plane 
XZ,  it  will  be  seen,  with  a  little  attention,  that  it  has  a  general  ap- 
plication. For,  after  having  reduced  all  the  forces  that  meet  this 
assumed  plane  to  two,  we  may  conceive  this  plane,  without  ceasing 
to  meet  diese  two  resultants,  so  placed  as  to  coincide  with  the  di- 
rections of  those  that  were  at  first  parallel  to  it ;  and  the  given 
forces  must  be  either  parallel  to  the  assumed  plane,  or  such  as 
being  produced  will  meet  it. 

72.  With  respect,  therefore,  to  forces  exerted  in  different 
planes,  the  result  is  not  the  same  as  that  with  respect  to  forces 
whose  directions  are  in  the  same  plane.     The  latter  forces,  as  we 

70.  have  seen,  may  always  be  reduced  to  a  single  one.  The  former 
are  reduced  to  two,  which  do  not  admit  of  being  represented  by 
one,  except  in  the  case  where  the  resultant  of  the  forces  which  act 
in  the  plane  XZ,  happens  to  meet  the  resultant  of  the  forces  per- 
pendicular to  this  plane. 

73.  We  can  accordingly,  in  the  manner  above  explained,  find 
the  two  resultants  of  any  number  of  forces  directed  in  different 
planes.  But,  although  the  method  here  pursued  may  be  useful  in 
certain  cases,  there  are  many  in  which  it  is  not  the  most  convenient. 
We  proceed,  therefore,  to  make  known  another. 

Fig.  25.  Let  p  be  any  one  of  the  proposed  forces,  and  AB  the  line  which 
represents  it.  From  any  fixed  point  X  draw  the  three  straight 
lines  XZ,  XY,  XT,  perpendicular  each  to  the  plane  of  the  other 
two.  These  mutually  perpendicular  lines  are  called  rectangular 
co-ordinates.  If  now,  upon  AB  as  a  diagonal,  we  form  the  rectan- 
gular parallelogram  ADBC,  having  its  plane  perpendicular  to  the 
plane  YXT,  and  its  side  BC  parallel  to  XZ;  and  then  upon  BD 
as  a  diagonal  we  form  the  rectangular  parallelogram  DFBE,  having 
its  plane  parallel  to  the  plane  YXT,  and  its  sides  BF,  BE,  parallel 
to  the  straight  lines  XT,  XY,  respectively  ;  it  is  evident  1.  That 
for  the  force  AB  we  may  substitute  the  force  5  C  parallel  to  XZ, 
or  perpendicular  to  the  plane  YXT,  and  the  force  BD  parallel  to 
this  latter  plane  ;  2.  That  for  the  force  BD  we  may  substitute  the 
force  BE  parallel  to  XY,  or  perpendicular  to  the  plane  ZXT,  and 


Centre  of  Gravity.  43 

the  force  jBF  parallel  to  XT,  or  perpendicular  to  the  plane  ZXY; 
so  that  the  force  p,  or  AB,  is  decomposed  into  three  forces  parallel 
to  three  rectangular  co-ordinates,  or  (which  is  the  same  thing)  into 
three  forces  perpendicular  to  three  mutually  perpendicular  planes. 

Now  what  has  been  said  of  the  force  p  is  evidently  applicable 
to  any  other  force  not  perpendicular  to  one  of  the  three  planes. 
If  therefore  all  the  forces  like  p,  be  considered  as  thus  decom- 
posed ;  and  we  afterwards  reduce  to  a  single  force  all  the  forces 
perpendicular  to  the  plane  ZXT,  the  same  thing  being  done  with  69. 
respect  to  all  the  forces  perpendicular  to  the  plane  ZXY,  and  also 
with  respect  to  all  the  forces  perpendicular  to  YXT,  it  will  be  seen 
that  we  may  reduce  any  number  of  forces,  directed  in  different 
planes,  to  three  forces  perpendicular  to  three  planes,  these  planes 
being  perpendicular  respectively  to  each  other. 

If  either  of  the  given  forces  happen  to  be  parallel  to  one  of 
the  straight  lines  XZ,  XY,  XT,  its  components  parallel  to  the  two 
other  straight  lines,  as  obtained  by  the  above  method,  would  be 
each  equal  to  zero. 

Such  are  the  general  principles  of  the  composition  and  decom- 
position of  forces. 


Of  the  Centre  of  Gravity. 

74.  Before  we  proceed  to  treat  of  the  particular  effects  pro- 
duced by  the  forces  whose  general  properties  we  have  been  con- 
sidering, it  is  necessary  to  speak  of  the  centre  of  gravity,  a  subject 
of  the  greatest  importance  in  all  inquiries  relating  to  the  motion  of 
machines  and  bodies  of  a  known  structure. 

By  gravity,  we  mean  the  force  which  urges  bodies  downward 
in  vertical  lines,  or  directions  perpendicular  to  the  surface  of  tran- 
quil waters.  If  the  earth,  or  the  surface  of  these  waters,  were 
perfectly  spherical,  the  direotions  of  gravity  would  all  meet  at  the 
centre.  But  although  this  surface  is  not  perfectly  spherical,  the  de- 
viation is  so  inconsiderable,  that,  for  the  objects  we  have  more 
immediately  in  view,  we  may,  without  sensible  error,  regard  the 
directions  of  gravity  as  meeting  at  the  centre  of  the  earth. 

The  earth  being  considered  as  a  sphere,  its  radius  is  estimat- 
ed at  3956  miles,  or  20887680  feet.     From  this  it  may  be  read- 


44  Statics. 

ily  inferred,  that  it  would  require  at  the  surface  of  the  earth  an 
extent  of  about  100  feet  to  subtend  an  angle. of  one  second  at  the 
centre.  Thus,  with  respect  to  a  machine  of  100  feet  in  length, 
the  directions  of  gravity  at  the  extreniities  would  want  only  one 
second  of  being  parallel.  Hence,  on  account  of  the  great  distance 
of  the  centre  of  the  earth,  compared  with  the  dimensions  of  any 
machine,  7ve  may  regard  the  directions  of  gravity  as  imrallel.  We 
may  also  for  the  same  reason,  consider  the  force  of  gravity,  exerted 
upon  different  parts  of  the  same  body,  as  the  same  in  point  of  magni- 
tude, and  capable  of  giving  the  same  velocity  in  the  same  time. 

75.  By  the  centre  of  gravity  of  a  body,  or  system  o(  bodies  (that 
is,  any  assemblage  whatever  of  bodies),  we  mean  that  point  through 
which  passes  the  resultant  of  all  the  particular  forces  exerted  by  the 
gravity  of  the  several  parts  of  the  body,  or  system  of  bodies,  in 
whatever  position  the  body  or  system  is  placed. 
Fig.  26.  If,  for  example,  in  the  actual  position  of  the  triangle  ABC,  the 
resultant  force  of  all  the  actions  of  gravity,  upon  the  several  parts  of 
this  triangle,  passed  through  a  certain  point  G  of  its  surface,  and  in 
another  position  a  b  c,  it  should  pass  through  the  same  point  G, 
this  point  is  what  we  call  the  centre  of  gravity.  We  shall  see  here- 
after that  the  resultant  in  question  passes  through  the  same  point  in 
all  possible  positions  of  the  given  body. 

7G.  The  centre  of   gravity  is  easily  determined  by  means  of 
what  we  have  said  upon  the  use  of  moments  in  finding  the  resul- 
61.     tant  of  several  parallel  forces. 

Fig.  27.  Let  there  be  any  number  of  bodies  m,  n,  o,  whose  masses  we 
will  consider  for  the  present  as  concentrated  in  the  points  A,  B, 
C,  situated  in  the  same  plane.  Let  u  be  the  velocity  which  gravity 
tends  to  give  to  each  in  an  instant ; 

u  X  m,  u  X  n,  u  X  o,  or  u  m,  u  n,  u  o, 
will  be  the  quantities  of  motion,  or  forces,  with  which  the  bodies  tend 
to  move  according  to  the  parallel  directions  A" A,  B"B,  C"C.  In 
order,  therefore,  to  find  the  position  of  the  resultant,  we  take  the 
sum  of  the  moments  with  regard  to  any  point  F,  assumed  at  pleas- 
ure, in  a  line  perpendicular  to  the  directions  of  the  forces,  and 
divide  this  sum  by  the  sum  of  the  forces;  we  have  therefore  for  the 
66.    value  of  the  distance  FG",  at  which  this  resultant  passes, 

pf^„  _  umX  FA"  +un  X  FB"  -\- u  o  X  FC" 
um-\-un-\-uo  '' 


Centre  of  Gravity.  45 

or,  by  suppressing  the  common  factor  ti, 

p^„  _  mXFA"-^nX  FB"  +  o  X  FC" ^ 
m  -\-  n  -\-  0 

In  like  manner,  if  we  draw  the  lines  AA',  BB',  CO,  parallel  to 

FG",  and  terminating  in  the  vertical   FO  ;  and  suppose  moreover, 

that  the  point  G,  taken  in  the  direction  of  the  resultant,  is  the  centre 

of  gravity  sought,  by  drawing  G'G  also  parallel  to  FG",  we  shall 

have 

FG"  =  G'G,     FB"  =  B'B,      FA"  =  A' A,      FC"  =  OC; 

whence 

^.^        m  X  A  A  +  n  X  BE  +  o  X  OC 
Cr'tr  =  j j , 

m  -\-  n  -\-  0 

that  is  (by  considering  the  masses  m,  w,  o,  as  representing  the  for- 
ces, the  velocity  u  being  common),  the  distance  of  the  centre  of 
gravity  of  several  bodies  from  an  assumed  straight  line,  is  found  by 
dividing  the  sum  of  the  moments  of  these  bodies  (taken  with  respect 
to  this  line)  by  the  sum  of  the  masses. 

Let  us  now  conceive  the  system  of  bodies  m,  n,  o,  reversed  in 
such  a  manner  that  FA",  instead  of  being  horizontal,  shall  become 
vertical,  &ic. ;  it  is  apparent,  that  in  order  to  find  the  distance  of  the 
resultant  from  the  line  FA",  now  vertical,  it  will  be  necessary  to 
take  the  sum  of  the  moments  with  respect  to  FA",  and  to  divide 
this  sum  by  the  sum  of  the  masses  ;  which  gives 

G"G  =  ""  ^  ^"^  +  "  ^  ^'^  "^ ""  ^  ^^^. 

?n  -\-  n  -{-  0 

Having   found  the  distance  of  the  point  G  from   two  fixed   known 
lines  FA",  FO,    the   position  of  this  centre    G  is  evidently  de-  Top.  i. 
termined. 

It  is  here  taken  for  granted  that  the  distances  A' A,  A" A,  B'B, 
BB",  he,  are  known,  since  the  point  through  which  FA",  FO, 
are  drawn,  is  assumed  at  pleasure. 

77.  If  the  distances  A" A,  B"B,  &c.,  are  each  zero  ;  that  is, 
if  all  the  bodies  are  in  the  same  straight  line  FA",  the  sum  of  the 
moments  with  respect  to  this  line  is  zero;  the  distance  G"G  is 
therefore  zero.  Accordingly,  if  several  bodies,  considered  as  points, 
are  in  the  same  straight  line,  their  common  centre  of  gravity  is  also 
in  this  line. 


46  Statics. 

78.  If  the  lines  FA",  FO,  are  either  of  them  drawn  in  such  a 
manner  as  to  have  bodies  situated  on  each  side  of  it  instead  of  the 
sum  of  the  moments,  we  should  say  the  sum  of  ihe  moments  that  are 
found  on  one  side,  minus  the  sum  of  the  moments  that  are  found  on 
the  other  side.  As  to  the  denominator  of  the  fraction  which  ex- 
presses the  distance  of  the  centre  of  gravity,  it  will  always  be  com- 
posed of  the  sum  of  the  masses,  since  all  the  forces,  by  the  nature  of 
gravity,  act  in  the  same  direction.  What  is  here  said  is  applicable 
to  any  number  of  bodies,  which,  being  considered  as  points,  are  sit- 
uated in  the  saiTie  plane. 

The  lines  FA",  FO,    are  called  the  axes  of  the  moments. 

79.  If  now  we  suppose  the  point  F,  which  we  at  first  took  arbi- 
trarily, to  be  in  G,  G'G  and  G"G  become  each  equal  to  zero. 
Therefore  the  sum  of  the  moments  with  respect  to  FA",  and  the 
sum  of  the  moments  with  respect  to  FO,  must  in  this  case  be  each 
equal  to  zero. 

Fiff  28  ^^'  ^^  "*^^  ^^)''  ^^^^^  ^^  ^^^^  ^""^  °^  ^^®  moments  of  several 
bodies  with  respect  to  the  straight  line  TS,  passing  through  the  point 
G,  is  equal  to  zero ;  and  the  sum  of  the  moments  with  respect  to 
the  straight  line  DE,  perpendicular  to  TS,  and  passing  also  through 
G,  is  in  like  manner  equal  to  zero ;  the  sum  of  the  moments  with 
respect  to  any  other  straight  line  LH,  passing  through  the  same 
point  G,  will  also  be  equal  to  zero. 

Indeed,  having  let  fall  upon  the  lines  DE,  TS,  LH,  the  per- 
pendiculars AA',  AA",  AA"';  if  we  suppose  that  the  point  1  is 
that  in  which  AA'  meets  LH,  from  the  right-angled  triangle  GA'l, 
we  have 

sin  GIA'  :   GA'  ::    sin  DGL  :  A'l, 
or 

A  A"  sin  DGL 


cos  BGL  :  AA"  :  :  sin  DGL  :  A'l  = 

e 

Al  =  AA'  —  A'l  =  AA' 


cos  VGL 

whence 

AA"  sin  DGL 


cos  DGL 

Now  from  the  right-angled  triangle   lAA'",  we  have,  radius  being 
supposed  equal  to  I, 

1  :  ^7  :  :  sin  AIA"  :  AA  '" 

:  :  cos  DGL  :  AA'"  =  AI X  cos  DGL; 


Centre  of  Gravity.  47 

that  is,  substituting  for  Al  its  value  above  found, 

AA'"  =  AA'  cos  DGL  —  AA"  x  sin  BGL ; 
hence,  if  we  multiply  by  the  mass  m  to  obtain  the  moment,  we  shall 
have 

m  X  AA"  =  m  X  AA'  x  cos  DGL  —  m  x  AA"  x  sin  DGL ; 
in  other  words,  the  moment  of  the  body  m  with  respect  to  the  axis 
LH,  is  equal  to  the  cosine  of  the  angle  DGL,  multiplied  by  the 
sum  of  the  moments  with  respect  to  the  axis  DE,  minus  the  sine 
of  the  same  angle  DGL,  multiplied  by  the  sum  of  the  moments 
with  respect  to  the  axis  TS. 

Now  it  is  manifest,  that  with  regard  to  any  other  body  n,  we 
should  arrive  at  a  similar  result,  with  the  exception  only  of  the  signs 
according  to  which  the  bodies  are  on  the  same  or  on  different  sides 
of  LH.  Consequently,  if  we  take  the  sum  of  all  the  moments  with 
respect  to  the  axis  LH^we  shall  find  that  it  is  equal  to  the  cosine 
of  the  angle  DGL,  multiplied  by  the  sum  of  the  moments  with 
respect  to  DE,  minus  the  sine  of  the  angle  DGL,  multiplied  by 
the  sum  of  the  moments  with  respect  to  TS.  But  each  of  these 
two  last  sums  is  by  supposiiion  equal  to  zero;  consequently  their 
products  by  the  cosine  and  sine  respectively  of  the  angle  DGL, 
will  be  each  equal  to  zero ;  therefore,  also,  the  sum  of  the  moments 
with  respect  to  any  axis  whatever  LH,  ivhich  passes  through  the 
centre  of  gravity  G,  is  equal  to  zero. 

81.  Hence  we  infer  that  the  resultant  action  of  all  the  partic- 
ular actions  of  gravity,  which  are  exerted  upon  the  several  parts 
of  a  system  of  bodies,  passes  always  through  the  same  point  of  this 
system,  whatever  be  its  position ;  for  it  is  not  with  respect  to  the 
direction  of  the  resultant  that  the  sum  of  the  moments  of  the 
several  parallel  forces  may  be  equal  to  zero.  68. 

Moreover,  although  the  inquiry  hitherto  has  been  only  respect- 
ing bodies  whose  centres  of  gravity  are  in  the  same  plane,  the 
method  is  not  the  less  applicable  to  the  case  where  the  parts  of  the 
system  are  in  different  planes. 

82.  If  the  bodies,  still  regarded  as  points,  are  not   in  the  same 
plane,  let  us  imagine  a  horizontal  plane  XZ,  and  from  each  of  the  Fig.  23. 
gravitating   points  p,  q,  r,  let  the  vertical   lines  A  p,  B  q,  C  r,  be 
supposed  to  be  drawn ;  and  in  order  to  determine  the  point  E, 


48  Statics, 

through  which  passes  the  resultant  q  E,  in  the  direction  of  which 
must  be  the  centre  of  gravity,  we  lake  the  moments  with  respect  to 
two  fixed  lines  FX,  FZ,  assumed  in  the  horizontal  plane,  perpen- 
dicular to  each  other ;  we  take,  I  say,  the  sum  of  the  moments,  as 
if  the  bodies  were  all  in  this  horizontal  plane  ;  and  having  divided 
each  of  the  two  sums  of  moments  by  the  sum  of  the  masses  or 
forces  p,  q,  r,  we  shall  have  the  two  distances  E'E,  E"E.  It 
will  only  remain,  therefore,  to  find  at  what  distance  EG,  below  the 
horizontal  plane  this  centre  is  situated. 

Now  if  we  imagine  the  figure  reversed,  the  plane  X.Z  becoming 
vertical,  and  2^  horizontal,  it  will  be  seen  that  in  order  to  deter- 
mine the  distance  E'G',  corresponding  and  equal  to  EG,  the 
distance  sought,  it  is  necessary,  according  to  the  method  above  pur- 
sued, to  take  the  sum  of  the  moments  with  respect  to  ZF,  as  if 
the  bodies  were  all  in  the  plane  ZV^,  and  to  divide  this  sum  by 
the  sum  of  the  masses  ;  we  have  then  every  thing  that  is  requisite 
in   order  to  fix  the  position  of  the  centre  of  gravity. 

83.  Hence,  by  recapitulating  v.'hat  we  have  said,  this  problem 
reduces  itself  to  the  following  particulars  ; 

(1.)  When  the  several  bodies,  considered  as  points,   are  situated 

Fig.  29.  in  the  same  straight  line,  we  take  the  sum  of  the  moments  with 

respect  to  a   fixed   point  JP,   assumed   arbitrarily  in  this  line,  and 

divide  this  sum  by  the  sum  of  the  masses,  and  the  quotient  will  be 

the  distance  of  the  centre  of  gravity  G  from  the  point  F. 

(2.)  When  the  several  bodies,  considered  as  points,  are  all  in  the 
YW.  27.  same  plane  ;  through  a  point  jP,  taken  arbitrarily  in  this  plane,  we 
suppose  two  lines  FA",  FC,  to  be  drawn  at  right  angles  to  each 
other  ;  and  having  let  fall  perpendiculars  upon  each  of  these  two 
lines  from  each  gravitating  point,  we  imagine  that  these  gravitating 
points  are  applied  successively  to  the  lines  FA",  FC,  where  their  per- 
pendiculars respectively  fall.  We  then  seek,  as  in  the  case  just  stated, 
what  would  be  the  centre  of  gravity  G"  in  FA",  and  what  would 
be  the  centre  of  gravity  G'  in  FC  ;  drawing  lastly  through  these 
two  points  the  lines  G"G,  G'G,  parallel  respectively  to  FC,  FA", 
and  their  point  of  meeting  G  will  be  the  centre  of  gravity  sought. 

(3.)  When    the   several    bodies,   considered  as  points,   are  in 
Fig.  23.  different  planes,  we  imagine  three  planes,  one  horizontal,  and  the 


Centre  of  Gravity.  49 

two  others  vertical  and  perpendicular  to  each  other.  From  each 
gravitating  point  we  suppose  perpendiculars  let  fall  upon  each  of 
these  three  planes  ;  we  then  take  the  sum  of  the  moments  with 
respect  to  each  plane,  and  dividing  each  of  these  sums  by  the  sum 
of  the  masses,  we  shall  have  the  three  distances  of  the  centre  of 
gravity  from  the  three  planes  respectively. 

84.  It  must  be  recollected,  moreover,  in  what  is  above  said, 
that  when  the  bodies  are  on  different  sides  of  the  line  or  plane  with 
respect  to  which  the  moments  are  considered,  it  is  necessary  to  take 
with  contrary  signs  the  moments  of  bodies  that  are  found  on  oppo- 
site sides. 

85.  We  will  here  make  a  remark,  that  Is  suggested  by  what 
has  been  said,  and  which  will  enable  us  to  abridge,  in  many  cases, 
the  process  of  finding  the  centre  of  gravity  as  well  as  the  solution  of 
other  problems. 

Since  the  distance  of  the  centre  of  gravity  is  expressed  by  the 
sum  of  the  moments  divided  by  the  sum  of  the  masses,  if  this  centre 
happen  to  be  in  the  point,  line,  or  plane,  with  respect  to  which  the 
moments  are  considered,  the  distance  being  zero,  the  sum  of  the 
moments  must  also  be  zero.  Therefore,  the  sum  of  the  moments 
with  respect  to  any  such  plane  as  may  pass  through  the  centre  of 
gravity  is  zero. 

86.  Hitherto  we  have  considered  bodies  as  so  many  points,  and 
we  have  seen  how  the  centre  of  gravity  of  all  these  points  may  be 
determined,  whatever  be  their  number  and  position.  Now  a  body 
of  any  size  or  figure  whatever,  being  only  an  assemblage  of  other 
bodies  or  material  parts,  which  may  be  considered  as  points,  it 
follows  that,  by  the  method  above  pursued,  we  may  determine  the 
centre  of  gravity  of  a  body  of  any  figure  whatever. 

Also,  since  the  centre  of  gravity  is  simply  the  point  through 
which  passes  the  resultant  of  all  d)e  particular  efforts  made  by  the 
several  parts  of  a  body  in  virtue  of  their  gravity,  and  since  this 
resultant  is  equal  to  the  sum  of  all  these  particular  efforts  ;  it  follows, 
that  we  may  in  all  cases  suppose  the  whole  weight  of  a  body  united 
at  its  centre  of  gravity,  and  the  weight  would  have  the  same  effect 
upon  this  point,  when  thus  united,  that  it  would  have  ia  its  actual 
state  of  distribution  through  all  parts  of  the  body. 
Mech.  "     7 


60  Statics, 

87.  When,  therefore,  it  is  proposed  to  find  the  common  centre 
of  gravity  of  several  masses  of  whatever  figure,  we  begin  by  seeking 
the  centre  of  gravity  of  each  of  these  masses,  which  is  attended  with 
no  difiicuhy.  Tiien,  the  weight  of  these  masses  being  considered  as 
united  each  at  its  centre  of  gravity,  we  seek  the  common  centre 
of  gravity,  as  if  all  these  bodies  were  points  situated  where  each 
has  its  particular  centre  of  gravity. 

SS.  Accordingly,  every  thing  which  we  have  said  hitherto  upon 
the  common  centre  of  gravity  of  several  bodies,  considered  as 
points,  is  equally  applicable  to  bodies  of  whatever  figure,  if  we 
take,  in  estimating  the  moments,  instead  of  the  distance  of  each 
body,  the  distance  of  its  particular  centre  of  gravity. 

89.  Hence,  finally,  if  several  bodies,  of  whatever  figure,  have 
their  particular  centres  of  gravity  in  the  same  straight  line,  or  in  the 
same  plane;  their  common  centre  of  gravity  ivill,  in  the  former  case, 
be  in  the  given  straight  line,  and  in  the  latter  in  the  given  plane. 


.Application  of  the  Principles  of  the  Centre  of  Gravity  to  particular 

Problems. 

Fig.  30.  90.  Let  AB  be  a  straight  line  uniformly  heavy.  It  will  be 
seen  at  once  without  the  aid  of  any  demonstration,  that  the  middle 
point  P,  of  its  length  will  be  its  centre  of  gravity.  But  in  order 
to  illustrate  and  confirm  the  theory  of  moments,  developed  in  the 
preceding  articles,  let  us  seek  the  centre  of  gravity  according  to  the 
principles  of  this  method. 

We  imagine  this  straight  line  divided  into  an  infinite  number  of 
points,  of  which  P  p  represents  one  ;  and  that  each  is  multiplied  by 
its  distance  from  a  fixed  point,  as  the  extremity  A  for  example. 
We  then  take  the  sum  of  these  products,  and  divide  it  by  the  sum 
of  the  parts  of  which  P  p  is  one,  that  is,  by  the  line  AB.  Ac- 
cordingly, if  we  call  AB,  a  ;  AP,  x ;  we  shall  have  Pp  =  d  x  ; 
Cal.  7.  and  the  moment  o(  P  p  will  be  equal  lo  x  d  x,  which  must  be  in- 
tegrated to   obtain  the  sum  of  the   moments.     This  sum  therefore 

Cat.  82.  will  be  equal  to  — ;  and  in  order  to  have  it  for  the  whole  extent  of 
the  line,  we  must  suppose  x  =  a,  which  gives  ^  for  the  entire  sum 


Centre  of  Gravity  in  particular  Bodies.  51 

of  the  moments.     Dividing  this  by  the  sum  a  of  the   masses,  we 

a^       a  . 

have  ?r-  or  p:  for  the  distance  of  the  centre  of  sravit}'  from  the  point      83. 

A.     Thus,  the  centre  of  gravity  of  a  straight  line,  uniformly  heavy, 
is  its  middle  point,  as  was  before  manifest. 

91.  Hence,   (I.)  In  order  to  have  the  centre  of  gravity  of  the  p-    ^i. 
perimeter  of  a  polygon,  it  is  necessary,  from  the   middle  of  each 

of  the  sides,  to  let  fall  perpendiculars  upon   two  fixed  lines  AB, 
AC,   taken   in   the   plane  of   this   polygon;    and,   considering   the 
weight  of  each  side  as  united  in  the  middle  of  this  side,  to  seek  the      86. 
common  centre  of  gravity  of  these  weights   in  the  manner  already 
explained. 

92.  (2.)  The  centre  of  gravity  of  th",  surface  of  a  parallelogram 
is  the  middle  point  of  the  line  which  joins  the  middle  of  two  opposite 
sides.      For,    by  considering    the    parallelogram    as    composed   of  Fig.  32. 
material   lines,  parallel  to  these  two  sides,  each  will  have  its  centre 

of  gravity  in  the  line  which  passes  through  the  middle  of  these  same 
sides.     The  common  centre  of  gravity,  therefore,  of  all   the   lines 
will  be  in  the  bisecting  line.     It  will,  moreover,    be   in   its   middle 
point,  since  this  line,  considered  as  sustaining  the  weights  of  all  the     90. 
other  lines,  is  uniformly  heavy. 

93.  (3.)  To  find  the  centre  of  gravity  of  a  triangle  ABC;  we  Fig.  33. 
draw  from  the  vertex  A  to  the  middle  D  of  the  opposite  side  BC, 

the  straight  line  DA,  and  from  the  point  D  we  take  DG  =.  \  DA. 

Indeed,  the  straight  line  DA,  which  divides  BC  into  two  equal 
parts  at  the  point  D,  divides  also  into  two  equal  parts  every  other 
line  LN,  parallel  to  jBC;  accordingly,  if  we  consider  the  surface 
of  the  triangle  as  an  assemblage  of  material  lines  parallel  to  BC, 
the  line  DA,  which  passes  through  the  particular  centres  of  gravity 
of  all  these  lines,  will  also  pass  through  tneir  common  centre  of 
gravity,  that  is,  through  the  centre  of  gravity  of  the  triangle.  For  77. 
the  same  reason,  the  line  CE,  which  passes  through  the  middle  ot 
AB,  will  in  like  manner  j)ass  through  the  centre  of  gravity  of  the 
triangle.  This  centre  is  consequently  at  the  point  of  intersection  G 
of  the  two  lines  CE  and  DA.  Now,  if  we  join  ED,  it  will  be 
parallel  to  AC,  since  it  divides  into  two  equal  parts  the   sides  AB, 


52  Statics. 

Geom.   ^C.     The  two  triangles  EGD,  AGC,  are  accordingly  similar,  as 

Geom.    well  as  the  triangles  ABC,  EBD  ;  we  have,  therefore, 

^°^'  DG   :  AG  ::  DE  :  AC  ::  BD  :  BC  ::   1    :  2', 

that  is,  DG  is  half  of -4 G,  and  therefore  one  third  o{  AIJ. 

94.  Hence,  in  order  to  find  the  centre  of  gravity  G  of  a  trape- 

Fig.  34.  zoid,  we  draw  KL  through  the  middle   points  of  the  two  parallel 

sides,  and  from  these   same   points  K,  L,  we  draw  the  lines  KA, 

LD,  to  the  vertices  of  the  opposite  angles  A,  D  ;  then  having 

taken 

KE=\KA,      LF=  ^  LD, 

we  join  EF,  which  will  cut  KL  in  G,  the  point  sought. 

For,  by  reasoning  as  we  have  done  in  the  case  of  the  triangle, 
we  shall  see  thai  the  centre  of  gravity  G  must  be  in  KL.  More- 
over, since  E,  F,  are  the   centres  of  gravity  respectively,  of  the 

93.  triangles  CAD,  ADB,  which  compose  the  trapezoid  ABDC,  the 
common  centre  of  gravity  of  the  two  triangles  or  of  the  trapezoid 

77.  must  be  in  EF;  it  follows,  therefore,  that  it  is  at  the  intersec- 
tion G. 

To  find  the  distance  LG,  which  we  shall  have  occasion  to  use 
hereafter,  w^e  draw  the  lines  EH,  FI,  parallel  respectively  to  AB  ; 
and  since 

KE=  ^  KA,     and    LF  z=  i  LD, 

we  shall  have 

EHz=  1  AL,     and     FI  =  I  KD, 

or 

EH=  I  AB,     and    FI  =  ^  CD. 

For  the  same  reason, 

KH=iKL,    LI=^KL', 
therefore 

HI=  ^KL. 

Now  the  similar  triangles  GHE,  GFI,  give 
EH  :    GH  ::  FI  :   GI; 
whence 

EH+  FI  :    GH+  GI    or     HI  : :  FI  :    GI; 

that  is, 

lAB  +  l  CD  :    '^KL   ::  iCD   :    GI; 


\ 


Centre  of  Gravity  in  particular  Bodies.  53 

therefore 

Gl 


^KL  X  I  CD  __  ^KL  X  CD 


l-AB-\-}  CD  ~    AB  +  CD    ' 
and,   because  LG  =  LI  -\-  GI,  if  we  substitute  for  LI  and   GI 
the  values  above  found,  we  shall  have 

=  7  ^^  X  jAB  +  2  CD) 
AB  +  CD 

We  remark  in  passing,  that  if  the  height  KL  of  the  trapezoid 
were  infinitely  small,  and  the  difference  of  the  two  sides  AB^  CD, 
were  infinitely  small,  these  sides  must  be  considered  as  equal,  so 

-  KL  X  3  AB 

that  the   distance  LG  would  reduce  itself  to ^   a  ri °^ 

2  AB 

i  KL ;  that  is,   the  centre  of  gravity  in  this  case  is  equally  distant 

from  the  two  opposite  bases. 

95.  To  find  the  centre  of  gravity  of  the  surface  of  any  polygon.  Fig.  35. 
we  divide  it  into  triangles,  and   having   found  the  centre  of  gravity 

of  each  triangle  in  the  manner  above  shown,  we  determine  the  com-     93. 
mon  centre  of  gravity  of  all  the  triangles,  by  considering  them  as 
so  many  masses  proportional  to  their  surfaces,  and  concentrated  each 
at  its  particular  centre  of  gravity,  agreeably  to  the  method  already    86. 
adopted. 

It  will  hence  be  seen  how  we  should  proceed  in  determining  the 
centre  of  gravity  of  the  surface  of  any  solid  figure  terminated  by 
plane  surfaces. 

96.  In  fine,  it  is  not  always  necessary  to  have  recourse  to  mo- 
ments in  finding  the  centre  of  gravity.  If  it  were  proposed,  for 
example,  to  determine  the  centre  of  gravity  of  the  perimeter  of  a 
regular  pentagon  ABCDE,  I  should  draw  from  the  vertex  of  one  Fig.  36. 
of  its  angles  A,  to  the  middle  of  its  opposite  side,  a  straight  line 
AH;  likewise  from  the  vertex  of  another  angle  E,  to  the  middle 

of  its  opposite  side,  a  straight  line  EI,  and   the  intersection  G  of 
these  lines  will  be  the  centre  of  gravity. 

Indeed  the  common  centre  of  gravity  of  the  two  sides  AB,  AE, 
is  evidently  the  middle  c,  of  the  line  b  a,  which  passes  through  their 
middle  points.     The  common  centre  of  gravity  of  the  two  sides 


54  Statics. 

BC,  DE,  is  for  the  same  reason  the  middle  of  the  line  IK  which 
passes  through  their  middle  points ;  and  the  side  CD  has  its  centre 
of  gravity  in  H.  Now  it  is  manifest  that  the  line  AH  passes 
through  the  middle  points  c,  e,  and  H ;  it  accordingly  passes  through 
the  common  centre  of  gravity  of  the  five  sides.  It  n)ay  be  shown, 
in  like  manner,  that  IE  also  passes  through  the  centre  of  gravity  ; 
therefore  this  centre  is  at  the  intersection  G  of  AH  and  IE. 

97.  By  pursuing  the  same  kind  of  reasoning  which  we  adopted 
in  the  case  of  the  triangle,  it  might  be  demonstrated  that  the  point 
G  is  the  centre  of  gravity  of  the  surface  of  a  regular  pentagon. 

In  general,  it  may  be  shown,  by  the  same  method,  that  the 
centre  of  gravity  of  the  perimeter,  as  well  as.  of  the  surface  of  any 
regular  polygon,  of  an  odd  number  of  sides,  is  the  point  of  in- 
tersection of  two  straight  lines,  each  of  which  is  drawn  from  the 
vertex  of  one  of  the  angles  to  the  middle  of  the  opposite  side. 
Fig.  37.  And  when  the  number  of  sides  is  even,  the  centre  of  gravity,  both 
of  the  perimeter  and  of  the  surface,  is  the  point  of  intersection  of 
two  straight  lines  drawn  through  the  middle  points  of  two  pairs  of 
opposite  sides.  We  might  also  extend  this  mode  of  reasoning  to 
the  circle,  by  regarding  it  as  a  polygon  of  an  infinite  number  of 
sides,  and  we  should  find  that  the  centre  of  gravity  of  the  circum- 
ference, and  of  the  surface,  is  the  centre. 

When  the  number  of  lines,  surfaces,  bodies,  &c.,  is  not  con- 
siderable, the  centre  of  gravity  may  be  found  by  the  method  of 
Fig.  38.  articles  53,  54.  Let  the  three  points  A,  B,  C,  for  example,  be 
the  centres  of  gravity  of  three  lines,  or  three  surfaces,  or  three 
bodies,  whose  weights  are  represented  by  the  masses  m,  n,  o. 
Having  joined  two  of  these  points,  as  B  and  C,  by  the  line  BC, 
we  divide  BC  at  G',  in  such  a  manner  as  to  give 

n  :  0   ::    CG'   :  BG' 
or 

n  -}-  0  :  n  :  :  CJ5  :  CG' ; 
and  the  point  G'  thus  found,  will  be  the  centre  of  gravity  of  the 
two  weights  n,  o.  We  now  draw  G'A,  and  supposing  the  two 
masses  n,  o,  united  in  G',  we  divide,  in  the  same  way,  G'A  in  the 
inverse  ratio  of  the  two  masses  m  and  n  -\-  o,  that  is,  so  as  to 
give 

n  +  o  :  m  ::  AG  '.  G'G, 


Centre  of  Gravity  in  particular  Bodies.  55 

or 

„  +  0  +  m  :  m  : :  AG'  :   G'G  ; 

and  the  point  G  will  be  the  common  centre  of  gravity  of  the  three 
weights  m,  n,  o.  We  might  proceed  in  a  similar  manner  with  a 
greater  number  of  bodies. 

98.  It  would  be  easy  to  deduce  from  what  precedes  an  easy 
method  of  finding  the  centre  of  gravity  of  the  surface  and  of  the 
solidity  of  any  cylinder  or  prism.  Indeed  it  is  evident  that  this 
centre  must  be  the  middle  of  the  line  that  passes  through  the  cen- 
tres of  gravity  of  the  two  opposite  bases,  since  bodies  of  this  form, 
being  composed  of  laminae  or  material  planes,  perfectly  equal  and 
similar  to  the  base,  may  be  considered  as  so  many  equal  weights 
uniformly  distributed  upon  this  line. 

99.  To  find  the  centre  of  gravity  G  of  a  triangular  pyramid 
SABC,  ive  draw  from  the  vertex  to  the  centre  of  gravity  F  of  the  Fig.  39, 
base,  the  straight  line  SF,  and  take  in  this  line,  reckoning  from  F, 

the  part  FG  ==  i  FS. 

To  show  that  G  is  the  centre  of  gravity  required,  from  the 
middle  D  of  the  side  AB,  we  draw  DC,  DS,  to  the  opposite  ver- 
tices C,  S,  of  the  pyramid,  and  having  taken  DF  =  ^  DC,  and 
DE  =  1  DS,  the  points  F,  E,  are  respectively  the  centres  of 
gravity  of  the  two  triangles  ABC,  ASB.  93: 

This  being  supposed,  if  we  consider  the  pyramid  as  composed 
of  material  planes,  parallel  to  ABC,  the  line  SF,  which  passes 
through  the  point  F,  of  the  base,  will  pass  through  a  point  similarly 
placed  in  each  of  the  parallel  planes  or  strata.     Thus  the  particular  Geom. 
centres  of  gravity  of  the   several   parallel  planes  will  all  be  in  the 
line  SF.     For  the  same  reason  the  particular  centres  of  gravity  of 
the  several   planes  parallel  to  ABS,  of  which  we  may  suppose  the 
pyramid   in  like  manner  composed,  are   all  in   the  line   EC.     Ac- 
cordingly, the  centre  of  gravity  of  the   pyramid   is  the   point   G, 
where  the  two  lines  SF,  EC,  situated  in  the  plane  SDC,  intersect 
each  other.     Now  if  we  draw  FE,  it  will   be  parallel  to  CS,  since 
DF  being  a  third  of  DC,  and  DE  a  third  of  DS,  these  two  lines 
are  cut  proportionally.     The  two  triangles  FEG,  GCS,  are  there- Ceom. 
fore  similar;  and  the  two  triangles  DFE,  DCS,  are  also  similar;*^®' 
whence 


56  Statics. 

FG    :    GS  ::   FE  :    CS  ::   DF  :   DC  ::   1    :   3; 

that  is,  FG  is  a  third  of  GS,  and  consequently  a  fourth  of  FS. 

100.  As  any  solid  may  be  decomposed  into  triangular  pyramids, 
knowing  the  centre  of  gravity  of  a  triangular  pyramid,  it  will  be 
easy,  by  the  method  of  moments,  to  find  the  centre  of  gravity  of  any 
body  whatever. 

101.  Such  is  the  general  manner  of  finding  the  centre  of  gravity 
of  bodies,  the  parts  of  which  are  independent  of  each  other,  or  rather 
when  we  have  not  the  expression  of  the  law  by  which  they  are  con- 
nected together. 

But  when  the  parts  of  a  figure  or  body  have  a  relation  that  can 
be  expressed  by  an  equation,  the  centre  of  gravity  may  be  found 
much  more  readily.  .■ —  *-;t-    f*^ 

Fiff  40  ^*^^'  ^^'^  ^^  ^^  required  to  find  the  centre  of  gravity  G,  of  any 

arc  of  a  curve  AM ;  we  imagine  an  infinitely  small  arc  M  m,  and 
take  for  the  axis  of  the  moments  any  line  CJV,  parallel  to  the  ordi- 
nates,  which  are  supposed  to  be  perpendicular  among  themselves. 
Suppose,  moreover,  that  the  distance  of  C  from  the  origin  ji  of  the 
abscissas  z=.  h,  h  being  taken  of  any  magnitude  at  pleasure.  To  ob- 
tain the  distance  GG'  of  the  centre  of  gravity  from  the  axis  CjY,  we 
^Q  must  take  the  sum  of  the  moments  of  the  arcs  Mm,  and  divide  it  by 
the  sum  of  the  arcs  Mm ;  that  is,  by  the  arc  AM.  Now  the  arc 
Mm  being  infinitely  small,  the  distance  of  its  middle  point  n,  from 
the  straight  line  CJV,  may  be  considered  as  equal  to  MJV.  We 
shall  have,  therefore. 

Mm  X   M.N' 
for  the  moment  of  this  infinitely   small   arc.     But,  calling  AP,  x, 

<Cal.   97. 


and  PM,  y,  we  shall  have  M  m  :=  \/d  x^  +  d  t/^,  and 
MJV=  PC  =h  —  x; 


iherefore  {h  —  x)  \/d  %-  -\-  d  y^  is  the   moment  of  the  arc  M  m ; 
and    consequently     P    (h  —  x)  \/d  x^  -j-  d  3/^,  or  the    integral  of 


{h  —  ar)  \/a  x^  -\-  d  y^,  is  the  sum  of  the  moments  of  all  the  infi- 
nitely small  arcs  M  m,  of  which  the  arc  AM  is  composed.  We 
have,  therefore, 


Centre  of  Gravity  in  particular  Bodies.  57 


A3I 

With  respect  to  the  arc  AM,   which  is  a  divisor    in    this    quantity, 

we  have  given  a  method  of  determining  it  exactly,  when   that  can  Cal.  96. 

be  done;   and  another  method  of  determining  it   by   approximation.  Cal. no. 

By    a  course  of  reasoning    similar    to    the    above,    we  should 
find  that  the  distance  GG"^  of  the  centre  of  gravity  from  the   axis 


•^^'■V  AM 


Such  are  the  general  formulas  which  serve  to  determine  the 
centre  of  gravity  of  any  arc  of  a  curve  of  which  we  have  the 
equation,  by  means  of  the  lines  designated  by  x  and  y. 

103.  If  the  arc  of  w^hich  we  wish  to  find  the  centre  of  gravity, 
is  composed  of  two  equal  and  similar  parts  AM,  AM'^  situated  pj„_  4j^ 
on  each  side  respectively  of  the  axis  of  the  abscissas,  it  is  evident 
that  the  centre  of  gravity  G,  will  be  in  the  straight  line  AP ;  we 
have,  therefore,  only  to  find  its  distance  from  the  point  C  Now 
it  is  plain  that  the  moments  of  the  two  arcs  M  m,  M'  m',  with 
respect  to  the  axis  JVW",  being  equal,  the  distance  CG  will  be 
equal  to 

2f{h~^x)\/dx^  +  dy^  76. 

31AJP 

For  example,  let  the  arc  MAM'  be  an  arc  of  a  circle ;  we  have  ^  . 

y  z=z  \/ax  —  X-,  a  being  the  diameter.     We  shall  easily  find,  andcai.  98. 
indeed  we  have  already  seen,  that 

/ ^  a  d  X 

Vd  x2  +  f/  »d  =    -" 

We  shall  have,  therefore, 

2r{h~-x)^dx^  +  dy^=   2/ia£-x)^x 
^  \  ax  —  1? 

/_  1 
(/t  —  x)  d  X  [a  X  —  X-)      ^' 

Supposing  now,  for  the  sake  of  greater  simplicity,  that  the   point  C 
is  the  centre,  then  AC  =:  h  =z  I  a;   we  shall  have,  therefore, 
Mech.  8 


Cal.  88. 


58  Statics. 

2  r{h  —  x)  ^dx^  +  di/  =  a  I*  {\  a  —  x)d  X  {ax—x^) 


=z  a  \^a  X  —  x^ ; 

an  integral  to  which  no  constant  is  to  be  added,  because  when 
a?  =  0,  this  integral  becomes  zero ;  as  indeed  it  ought,  since  the 
sum  of  the  moments  is  then  evidently  nothing. 

We  have,  therefore,  finally, 

2 j" (/i  _  cc)   y/dx^-{-dy^  —  a  A^ax—%% 
and  consequently, 

^^ MAW  ^^  MA31'  MAM      ' 

which  gives  this  proportion, 

MAM'  :  MM  :  :  CA  :  CG. 

Thus  we  obtain  the  following  rule  ;  that  the  distance  of  the  centre 
of  a  circle  from  the  centre  of  gravity  of  any  arc  of  this  circle,  is  a 
fourth  proportional  to  the  length  of  the  arc,  its  chord,  arid  radius. 

These  formulas  may  be  applied  to  any  other  curve. 

We  pass  now  to  the  consideration  of  the  centre  of  gravity  of 
plane  surfaces  bounded  by  curved  lines. 

Fig.  42.  104.  Let  it  be  required  to  find  the  centre  of  gravity  of  the  sur- 
face-4P.M;  and  let  G  represent  this  centre.  In  order  to  obtain 
the  distance  GG',  it  is  necessary  to  take  the  sum  of  the  moments  of 
the  small  trapezoids  MP  p  m,  with  respect  to  CJV,  and  to  divide 
this  sum  by  the  sum  of  the  trapezoids,  that  is,  by  the  surface  APM^ 
Now  the  centre  of  gravity  F  of  this  small  trapezoid  must  be  in  the 
middle  point  of  the  straight  line  n  K,  equally  distant  from  MP  and 
wp,  which  point  we  can  suppose  to  be  in  MP,  on  account  of  the 
infinitely  small  height  F  p.  We  shall  have,  therefore,  FL  =  CP ; 
and  the  moment  oi  P  p  m  M  will  be 

P  p  m  M  X   CP, 
that  is,  {h  —  x)y  dx,  calling  always  CA,  h,  and  AP,  x.      There- 
fore the  sum  of  the  moments  will  he  J*  {h  — x)  y  d  x,  and  conse- 
quently the  distance 

„,  _f{h—x)  ydx 
^^   -        AFM         ' 


Centre  of  Gravity  in  particular  Bodies.  59 

It  will  be  found,  likewise,  that  the  distance 

^^    -     APM   ' 

105.  The  centre  of  any  plane  surface  may  be  found,  in  the 
same  manner,  by  decomposing  it  into  infinitely  small  trapezoids. 

For  example,  let  the  surface  in  question  be  the  triangle  AJVJV', 
and  take  the  base  JVJV'  and  the  height  AC  for  the  axes  of  the  mo- 
ments;   now   calling  AP,x,  MM',  y,   and  AC,h,  we  shall   have  ^'^"  '^^' 
MM'  m'  m  =z  y  dx  ;  and  the  moment  of  this  trapezoid,  with  respect 
to  JVC,  will  be  (A  —  x)  y  d  x.     Therefore,  the  distance  GG'  of  the 

centre  of  gravity  from  the  base,  will  be  a  mm'  '    ^°^^  *^^^^' 

ing  c  the  base,  we  have 

AC  :  AP  ::  JVJV'  :  MM'; 
that  is, 

,  ex 

h  :  X  ::  c  :  y  = -j  J 

therefore/ (A  —  x)  y  d  x  becomes/ (A  — x)  — r— ,  or 

f  c 

•y  [hxdx  —  a;2  rf  x), 

c     /h  a;2        3;3\  f.  3;2 

the  value  of  which  's  -r-  (  -^ q  )  ^^  aX  (.^  ^  —  2  x).     Now 

the   surface  AMM'  is ^—^ —  or   ^-t  >   therefore  the  distance 

GG'  of  the  centre  of  gravity  of  the  surface  from  the  base  is 


'^{Sh-2x) 

^A___ ori(3A-2^), 

which,  when  x  =  h,  becomes  \  h,  to  which  this  distance  is  therefore 
equal.  Now  if  we  draw  the  line  AGL,  the  similar  triangles  ACL, 
GG'L,  give 

LG  :  LA  ::  GG'  :  AC  ::  ^h  :  h  ::  \   :  3 

therefore  LG  =  \  LA,  which  agrees  with  what  has  been  before    9S. 
demonstrated. 


CO  Statics. 

106.  Let  us  now  apply  the  formulas  to  curved  lines.     Suppose 
that  APM  is  a  portion  of  a  circle  whose  diameter  is   a,  and  whose 
Fig.  44.  centre  is  C;    we  have  then  h  =z  AC  =.  \  a.     Now 


y  =  \/ax—  x^; 
the  quantity  /  (A  —  x)  y  d  x,  hecomes  therefore 


Cal.  8S.  f{^a—x)dxj^ax—  x^, 

or /(I  a — x)dx[ax  —  a;-)-,    wliich    is    an    integrable  quantity, 

and  being  integrated,  gives  \  [a  x  —  x^)'^  ;  a  quantity  to  which  no 
constant  is  to  be  added,  because  it  becomes  zero,  when  x^=.  0;  as 
it  evidently  ought.     We  have,  therefore, 

"^^   ~        APM         -ATM' 

With  respect  to  GG",  since  y  =  a^/u.  x  —  x^,  we  shall  have, 


104.  GG' 


_  fi-  (a  x  —  x^)  dx 


APM  ' 

but  r  ^  {a  X  —  x^)  d  X,  or  /*^-  {a  x  d  x  —  x^  d  x), 


^Q^-^^)^'r\^'i^ci-2x); 


we  have,  therefore, 

j-,  x2  (3  a  _  2  x) 
^^    = APM • 

If  the  question  related  to  the  entire  segment,  since  it  is  evident  that 

Fig.  45.  the  centre  of  gravity  E,  must  be  in  the  radius  CA,  which  bisects  the 

arc,  and  that  it  must   be  at    the    same  distance    from   JYJV^  as  the 

centre  of  gravity  of  each  of  the  two  semi-segments  APM,  APM', 

we  shall  have 

CF  —  i^  —  gV  X  8  X  PJfi  __  ^MM'^ 
~  APM  ~~  APM         ~~    APM 

~  ^^~M  ~  ~AMW  ' 

that  is,  the  distance  of  the  centre  of  a  circle  from  the  centre  of  grav- 
ity of  any  one  of  its  segments,  is  equal  to  the  twelfth  part  of  the  cube 
of  the  chord,  divided  by  the  surface  of  this  segment. 

Fig.  45.       107.  The  centre  of  gravity  of  a  sector    CMAM'  may  be  easily 
found,  by  observing  that  £,  the  centre  of  gravity  of  the   segment 


93. 


Centre  of  Gravity  in  particular  Bodies.  61 

•MAM',  G  that  of  the  sector,  and  F  that  of  the  triangle,  are  all  in 
the  radius  CA  ;  and  that,  according  to  the  principle  of  moments,  the 
moment  of  the  sector  must  be  equal  to  the  moment  of  the  segment 
plus  that  of  the  triangle.     We  have  then 

CMAM'  X  CO  =  MAM  x  CE  +  CMM  X  CF. 

JL  JO  i|3 

Now  we  have  just  found  CE  =.   ^    '   y,  which  may  be  changed 

*'"^°  ¥-rFJl=MXW'  therefore  CE  X  MAM  =  |  pJi^. 
We  know,  moreover,  that  CMM'  =  PM  X   CP,  and  that 

CF  =  ^  CP, 

so  that  CMM'  X  CF  is  reduced  to  |  PM  X  CP^-  Substituting, 
therefore,  these  values,  we  have 

CMAM'  X  CG  =  iWf  -{-  ^  PM  X  cP" 

=  I  PM  {PJI^  +  CF^) 

=  I  PM  X  VM^,  on  account  of  the  right-angled  triangle  CPM ; 
consequently, 

CiUAM'      ' 
But  the  surface  of  the  sector  CMAM,  is  equal  to  the  arc  MAM 

CM  r 

multiplied  by  -^,  therefore  ^q°™' 

Cr  —  4  P^J^  X  c372  _  ^PMx  CM  __  I  MM'  X  CA 
~  MAM  X  CM  ~        MAM'        ~         MA3I'       ' 

2 

That  IS,  the  distance  of  the  centre  of  a  circle  from  the  centre  of 
gravity  of  any  one  of  its  sectors,  is  a  fourth  proportional  to  the  arc, 
the  radius,  and  two  thirds  of  the  chord  of  the  arc. 

The  formulas  above  found  may  be  applied  to   any  other  curve, 
as  the  parabola,  &ic. 

lOS.  We  now  proceed  to  the  consideration  of  curved  surfaces, 
confining  ourselves  to  those  of  solids  of  revolution.  Reasoning,  then, 
as  in  the  preceding  articles,  it  will  be  perceived  that  the  centre  of 
gravity  of  each  elementary  zone,  is  in  the  axis  of  revolution  CA,  and  Fig.  46. 
that  it  must  be  regarded  as  at  the  centre  P  of  one  of  the  bases  of 
this  zone,  considered  as  having  an  infinitely  small  breadth.  But  we 
have  seen  that  the  expression  for  this  zone  \s  2  n  y  j^d  aP  -\-  d  y%  Cal.  97. 


62  Statics. 

n  representing  the  ratio  of  the  diameter  to  the  circumference.  We 
shall  have,  therefore,  (denoting  always  by  h,  the  distance  AC  o{  A, 
the  origin  of  the  abscissas,  from  AW',  the  axis  of  the  moments)  2  n 
(h  —  x)  y  \/d  x^  4"  ^  y^  ^oi"  ^'^^  moment  of  this  zone  ;  from  which 
it  follows  that  the  distance  of  G,  the  centre  of  gravity  of  the  sur- 
face, from  the  point  C,  designating  this  surface  by  a,  will  be 


f  2n{h—x)  y  yrf  x^  +  d  y^ 
a 

109.  Let  us  suppose,  in  order  to   apply  this  formula,  that  it  is 

proposed  to  find  the  centre  of  gravity  of  the  convex  surface  of  the 

F«g.47.  right  cone  AJV^f' ;  we  denote  AP  by  x,  PM  by  y,  the  height  AC 

by  h,  CJV  the  radius  of  the  base  by  a,  and  the  side  AJY  by  e.     On 

account  of  the  similar  triangles,  ACJV,  M  r  m,  we  have 

AC  :   AN  :  :   Mr  :   Mm-, 

that  is, 

e  d  X 


h  :  e  ::  d  X  :  \/d  x^  -\-  d  y^  =.  — - 

We  have  also,  on  account  of  the  similar  triangles,  w3  CJV  and  APM, 
AC   :    CJV  ::  AP  :  PM, 

that  is. 


a  X 


therefore 


f  2  7t  (h  —  x)  y  A^dx^-{-df 
becomes  J  2  «  X  (A -a:)  X  ^  X  ^^j 

or  - — ^ —  (hx  dx  —  x^d  x)  ; 

of  which  the  integral  is 

2  71  a  e  /h  x^        x^\  n  a  e  x^  ..  ,        _     . 

Geom.  Now,  the  surface  of  the  portion  AM'LMA,  or  a,  is  equal  to 

AM 

—^  X  circum  PM,  and  we  have 

AC  :  AP  ::  AN  :  AM,  =  i^-ii^^; 
therefore. 


Centre  of  Gravity  in  particular  Bodies.  63 

AP  X  AN  ^^     .  „,,        XX  e       2iiax       naex^ 

"^  =       ^2  AC       ^  """'^  ^'^=   "27r  X  -T-=~W-  ' 

therefore  the   distance  of  the   centre   of  gravity  of    the  surface 
AM'LMA,  from  the  point  C  is 

.J±i-^ ,  or  1  (3  h-2x). 

n  a  e  x^ 


Therefore,  when  a;  =  ^,  or  AP  =  AC,  we  shall  have  the  dis- 
tance CG,  of  the  centre  of  gravity  of  the  whole  curved  surface  of 
the  cone, 

=  i{3h-2h)  =  lh; 
that  is,  the  centre  of  gravity  is  found  in  the  same  manner  as  that  of 
the  surface  of  the  triangle  AJVJV. 

110.  For  a  second  example,  we  take  the  sphere.     We  now  have  Fig.  48. 
y  =  Y/a  X  —  x^,  a  being  the  diameter,  and 

i  a  d  x 


\/dx'^  +dy^=   ~ ;  ;  Cal.  98. 

\/a  X  — x^ 


therefore  r  2  n  {h  —  x)  y  \/d  x^  -\-dy^  will  become 

J*  2  n  {h  —  x)  ^  a  d  X  'f 

and  this  expression,  C  being  the  centre,  which  gives  h=z  \  a,  will 
be  equal  to  f"  ti  a  {^  a  d  x  —  x  d  x),  which,  being  integrated,  is 
n  a  {^  a  X  —  I  a:^)  or  71  rt  X  Q  a  —  ^  x).  Now  we  have  found 
that  the  surface  a  of  the  spherical  segment  AMLM'A,  was  n  a  x)  Cai.  98. 
we  have,  therefore,  for  the  distance  CG  of  C,  the  centre  of  the 
sphere,  from  the  centre  of  gravity  G  = 

^  ( I  a  _  A  ^ )  =  i  a  -  1  X  =  C^  -  1  ^P ; 

that  is,  the  centre  of  gravity  G  is  in  the  middle  of  the  altitude  AP 
of  this  segment.  Hence  we  derive  the  general  conclusion,  that  the 
centre  of  gravity  of  the  surface  of  a  spherical  zone,  comprehended 
between  two  parallel  planes,  is  the  middle  point  of  the  altitude  of 
this  zone. 

111.  We  shall  terminate  this  branch  of  our  subject,  with  the  in- 
vestigation of  the  centres  of  gravity  of  solids. 


64  Statics. 

Fig.  46.  If  we  consider  a  solid  as  made  up  of  laminaB  infinitely  thin,  and 
parallel  to  each  other,  and  represent  generally  by  a  one  of  the  op- 
posite bases  of  each  lamina,  and  by  d  x  its  thickness,  we  shall  have 
or  rf  a;  as  the  expression  for  each  lamina ;  and  consequently 
a  [h  —  x)  d  X  for  its  moment  with  respect  to  a  plane  parallel  to  these 
laminae,  whose  distance  AC  from  the  vertex^  we  represent  by  h. 
Therefore,  denoting  by  h  the  bulk  ALM'MA,  the  distance   of  the 

centre  of  gravity  from  C  will  be  z=.-^-—^ — - — '- .         Now  the 

Cal.  100,  value  of  i  is  determined  by  methods  which  have  been  heretofore  giv- 
en, and  that  of  Ta  (A  —  x)  d  x  is  found  by  the  same  methods,  when 
the  value  of  a  is  known  in  terms  of  x.  We  shall  thus  obtain  the  dis- 
tance of  the  centre  of  gravity  from  a  known  plane.  We  might  find 
in  the  same  way  the  distance  of  this  centre  from  each  of  the  two 
other  planes,  perpendicular  to  one  another,  and  to  the  first ;  but  we 
shall  confine  ourselves  for  the  present  to  those  solids,  of  which  the 
parallel  laminae  have  their  respective  centres  of  gravity  all  in  the  same 
straight  line,  as  pyramids,  and  solids  of  revolution. 

112.  We  begin   with  pyramids.     Let  h  denote   the  height  ^C 

Fig.  49.  of  any  pyramid ;  x  the  perpendicular  distance  AP  of  any  lamina 

from  the  vertex  ;  c^  the   surfiice  of  the  base ;  we   shall  have  the 

surface  of  the  lamina  situated  at  the  distance  x  from  the  vertex,  by 

this  proportion 


Gcom. 
409. 


we  have,  therefore, 


>-2 


]fi    ' 


/t2      ' 

whence  J  a  (h  —  x)  d  x  becomes 

fi  ^''  ^'dx-x^  d  x)  =  g  Q^  -  0 

Geom.  But  the  pyramid  which  has  x  for  its  height,  and  for  its  base 
(J  or  -Tg-  is  =  ^  ;  the  distance  of  the  centre  of  gravity,  there- 
fore, is 


Centre  of  Gravity  in  particular  Bodies.  65 

or  1  (4  /»  —  3  a:) ; 

this  quantity,   when   x  ^=  h  =^  ^  C,  is  reduced  to  i  A,  and  we  have 

the  height  CG'^  of  the  centre  of  gravity  G,  above  the  base  =  J  A. 

Now  let  G'  represent  the  centre  of  gravity  of  the  base  ;  the  line 
jIG'  will  pass  through  G,  the  centre  of  gravity  of  the  pyramid  ;  and 
the  parallels  G^'G,   G'C,  will  give 

G"Cov  \h  :  ACovh  :-.   GG'  :  AG' ; 
whence  GG'  z=z  \  JIG' ',  which  confirms  what  we  have  before  said,     99- 
and  shows  that  the  centre  of  gravity  of  every  pyramid,  is  one  fourth 
of  the  distance  from  tlie  centre  of  gravity  of  the  base,  to  the  vertex. 

113.  With  respect  to  solids  of  revolution,  the  general  value  ofGeom. 
a  is  71^^;  the  general  expression   for  the  distance  of  the  centre  of " 

gravity  will  thus  be    - —      ^  , .     This    formula   may    be 

applied  to  the    sphere,   the   ellipsoid,  &c.,  and   by   means  of  the 
ellipsoid  may  be  determined  the  centre  of  gravity  of  masts. 

114.  What  we  have  said  with  respect  to  centres  of  gravity, 
will  enable  us  to  arrive  at  a  solution  in  any  case  that  may  occur ; 
we  shall,  notwithstanding,  point  out  particularly  tlie  course  to  be 
pursued  in  order  to  find  the  centre  of  gravity  of  the  immersed  part 
of  a  ship's  bottom,  or  rather  of  a  homogeneous  solid  of  this  form. 

We  may  suppose  the  centre  of  gravity  to  be  in  a  vertical  plane 
passing  through  the  axis  of  the  keel,  and  we  have  only  to  deter- 
mine its  horizontal  distance  from  a  vertical  line  drawn  through  a 
given  point  of  the  stern-post,  and  its  vertical  distance  from  the 
keel. 

For  each  of  these  objects  we  must  begin  by  determining  the 
centre  of  gravity  of  a  surface  AJVDFPB,  bounded  by  two  par- 
allel lines  AB,    DF,   and    two    equal   curves,   similar    to   AND,  Fig.  50. 
BPF. 

If  we  had  the  equation  of  this  curve,  nothing   would  be  more 
easy  than  to  determine  its  centre  of  gravity  G,  by  the   preceding 
methods.     But  not  having  it,  we   must  conceive  the  line  CE  to 
Mech.  9 


178. 


91. 


66  Statics. 

pass  ihrougli  C  and  E,  the  iniddle  points  of  AB,  DF,  respective- 
ly, and  that  this  line  is  divided  by  the  perpendiculars  TH,  KM, 
&z;c.,  into  equal  parts,  so  small  that  the  arcs  comprehended  be- 
tween any  two  adjacent  perpendiculars,  shall  not  differ  sensibly 
from  straight  lines.  We  must  next  take  the  moments  of  the  trape- 
zoids DTHF,  TKMH,  &tc.,  with  respect  to  the  point  E,  and 
Geom,  divide  the  sum  of  these  moments  by  the  sum  of  the  trapezoids, 
that  is,  by  the  surface  JINDFPB.  This  surface,  being  composed 
of  trapezoids,  is  readily  determined.  We  have,  therefore,  only 
to  find  a  simple  expression  for  the  sum  of  the  moments.  Now  the 
distance  of  the  centre  of  gravity  of  the  trapezoid  THFD,  from 
the  point  E,  is 

IIE  X  (DF  -f-  2  TH) , 
DF^^TH  ' 

that  of  the  trapezoid  TKMH  fiom  the  same  point  E,  will  be,  for 
the  same  reason,  taken  in  connexion  with  the  equality  of  the  lines 
IE,  IL,  &c., 

IIE  X  {TH-j-2  KM)        ,p        ^  J£  (4  TH -}- 5  KM) 
TH  -\-  KM  +       '  ^^  TH  -f  KM 

In  like  manner,  the  distance  of  the  centre  of  gravity  of  the 
trapezoid  NKMP,  will  be 

^lEx  (KM-^2NP)     ,cyTp,,  \  lEX  {7  KM  +  8  NP) 
KM  -\-  NP  "•"  '^       '  KM  -f-  NP  ' 

and  so  on. 

Geom.  Now  if  we  multiply  each  distance  by  the  surface  of  the  cor- 

responding trapezoid,  that  is,  by  half  the  sum  of  the  two  parallel 
sides,  multiplied  by  their  common  height  LE,  we  shall  have  for  the 
series  of  these  moments, 

^IE"  X  (DF  +  2  TH),     1  /E^  X  (4  TH  +  5  KM), 
}IE^  X  {7  KM+8  NP), 
and  so  on  ;  the  sum  of  which  will  be 

^m''  X  {DF  +Q  TH-{- 12  KM +18  NP  + 
24  qS  -{■  14  AB). 

It  may  be  observed,  that  if  there  were  a  greater  number  of 
divisions,  the  multiplier  of  the  last  term,  which  is  here  14,  would 
be  in  general  2  -j-  3  (n  —  2)  or  3  n  —  4,  n  representing  the 
whole  number  of  the  perpendiculars  DF,  TH,  he,  including  JlB, 


Centre  of  Gravity  in  particular  Bodies.  67 

which  may  be  zero.      So  that  the  general  expression  for  the  sum 
of  the  moments  is  reduced  to 

m  (1  DjP  +  Tif  +  2  KM  +  3  iVP  +  4  Q^  -}-  &c.  .  . 
+  (iiLZll)  AB). 

But  it  is  evident  that  the  surface  ANDFPB  has  for  its  ex- 
pression, 

IE  X  {iDF-{-  TH+  KM  +  iVP  +  &c.  .  .  +  i  AB)  ; 
and  hence  the  distance  of  the  centre  of  gravity  G,  namely, 

IEX{lDF-^TH  +  2 KM+SJ^P  +  &c.  .  .  +  ^^"-^)  ^B) 
IDF  -j-  TH+  KM+J^P^  &c.  .  .  +  i  ^5 

This  formula,  expressed  in  common  language,  furnishes  the 
following  rule  ; 

To  find  the  distance  of  the  centre  of  gravity  G,  from  one  of 
the  extreme  ordinates  DF, 

(1).  Take  a  sixth  of  the  first  ordinate  DF  ;  a  sixth  of  the  last 
ordinate  AB,  multiplied  by  triple  the  mimher  of  ordinates  less  4  ; 
then  the  second  ordinate,  double  the  third,  triple  the  fourth,  and  so 
on ;  which  may  be  called  the  first  sum,. 

(2).  To  half  the  entire  sum  of  the  two  extreme  ordinates,  add 
all  the  intermediate  ordinates,  for  a  second  sum  ; 

(3).  Divide  the  first  sum  by  the   second,  and  multiply  the  quo-     83. 
tient  by  the  common  interval  between  two  adjacent  ordinates. 

For  example,  if  there  were  7  perpendiculars,  whose  values 
were  18,  23,  28,  30,  30,  21,  0,  feet;  and  each  interval  were 
20  feet ;  I  should  take  a  sixth  of  18,  which  is  3  ;  and  since  the 
last  term  is  0,  I  should  add  to  3,  the  second  ordinate  23,  double 
of  28,  triple  of  30,  4  times  30,  and  so  on,  which  would  give  397. 
To  the  half  of  18,  I  should  next  add,  23,  28,  &.c.,  the  result  of 
which  would  be  141  ;  now  dividing  397  by  141,  and  multiplying 
by  20,  I  should  have 

— -r-, or  -^TTT^  =  56  feet  4  inches  nearly.* 

141  141 

When  we  once  know  how  to  determine  the  centre  of  gravity 
of  any   section  of  a  solid,  that  of  the  solid  itself  is  easily  found. 

*  See  Bouguer,  Traiie  du  Navire,  p.  213. 


68  Statics. 

Hence,  by  means  of  what  is  above  laid  down,  we  can  determine 
the  centre  of  gravity  of  the  hold  of  a  ship,  or  of  the  space  em- 
braced by  the  outer  surface  of  a  ship's  bottom.  Let  it  it  be  pro- 
posed to  find  the  distance  of  the  centre  of  gravity  of  ti)is  space 
from  tlie  keel.  We  imagine  it  composed  of  several  laminae  par- 
Fig.  5],  allel  to  the  section  at  the  water's  edge.  The  bullc  of  each  lam- 
ina will  be  equal  to  half  the  sum  of  the  two  opposite  surfaces  of 
Geom.  ijjjg  lamina,  multiplied  by  their  perpendicular  distance,  and  the 
centre  of  gravity  will  be  at  the  same  height  in  tliis  lamina,  as  in  the 
trapezoid  ABCD,  which  is  a  section  of  this  lamina,  made  by  a 
vertical  plane  passing  through  the  keel.  We  see,  therefore,  that 
the  reasoning  to  be  made  use  of  here,  in  order  to  find  the  height 
GE,  of  the  centre  of  gravity,  is  precisely  the  same  as  that  in  the 
last  case,  substituting  only  for  perpendicular  or  ordinate,  the  word 
section  ;  we  have,  therefore,  this  rule  ; 

(1).  Take  a  sixth  of  the  lowest  section;  a  sixth  of  the  highest, 
multiplied  by  triple  the  number  of  sections  less  4  ;  the  second  section 
from  the  lowest,  double  the  third,  triple  the  fourth,  and  so  on  ;  and 
call  the  result  the  first  sum. 

(2).  Take  half  the  sum  of  the  lowest  and  highest  sections,  and 
all  the  sections  betiveen  them,  for  the  second,  sum. 

83.  (3).  Divide  the  first  sum  by  the  second,  and  multiply  the  quo- 

tient by  the  common  distance  between  two  adjacent  sections. 

We  may  make  use  of  the  same  method  in  finding  the  distance 
Fig.  51.  of  the  centre  of  gravity  from  the  vertical  line  JiZ,  drawn  through 
a  determinate  point  B  of  the  stern-post,  by  imagining  the  bot- 
tom cut  by  planes  parallel  to  the  midship  frame  ;  but  as  it  would 
be  necessary  to  measure  the  surfaces  of  these  sections,  it  is  better 
to  make  use  of  those  which  have  been  already  measured,  in  the 
last  operation ;  accordingly  we  determine  by  the  above  method 
the  centres  of  gravity  G',  G',  of  the  several  sections  parallel  to  the 
keel.  Their  distances  from  the  vertical  XZ  will  be  each  the  same 
as  that  of  the  centre  of  gravity  G'  of  the  corresponding  lamina. 
We  now  multiply  each  section  by  the  distance  of  its  centre  of 
gravity  from  the  lines  XZ,  and  regarding  the  several  products  as 
the  ordinates  of  a  curved  line,  like  those  in  figure  50,  we  add  the 
half  sum  of  the  two  extreme  products,  to  the  sum  of  all  the 
mean  products,  and  divide  the  enfire  sum  by  the  sum  of  all  the 


Properties  of  the  Centre  of  Gravity.  69 

mean  sections,  plus  half  the  sum  of  the  two  extreme  sections,  the 
common  thickness  of  the  laminae  being  suppressed   as  a  common     83. 
factor  to  the  dividend  and  divisor. 

With  respect  to  the  centre  of  gravity  of  the  vessel  itself,  wheth- 
er laden  or  not,  the  investigation  cannot  be  reduced  to  so  simple  a 
process.  We  must  take  into  particular  consideration  the  differ- 
ent parts  which  compose  both  the  vessel  and  its  lading.  Having 
found  the  moments  of  these  different  parts  with  respect  to  a  hori- 
zontal plane,  supposed  to  pass  through  the  keel ;  and  the  moments 
with  respect  to  a  vertical  plane  taken  at  pleasure  perpendicular 
to  the  keel ;  we  divide  each  of  these  two  sums  by  the  whole  weight 
of  the  vessel,  and  we  obtain  the  height  of  the  centre  of  gravity, 
and  its  distance  from  the  vertical  plane  with  respect  to  which 
the  moments  were  considered  ;  and  as  it  must  also  be  in  the 
vertical  plane  which  passes  through  the  keel,  we  shall  have  its 
position.  But  it  may  be  remarked,  that  in  the  calculation  of  these 
moments,  we  must  multiply,  not  the  bulk  of  each  part,  but  its 
weight,  by  the  distance  of  the  centre  of  gravity  of  tliis  part ;  which 
centre  is  easily  determined  after  all  that  has  been  said  upon  this 
subject. 

Pro2)erties  of  the  Centre  of  Gravity. 

115.  It  is  evident  from  what  we  have  said  upon  the  subject  of 
the  centre  of  gravity,  and  upon  the  resultant  of  parallel  forces, 
that  if  the  parts  of  a  body  or  system  of  bodies  have  each  the 
same  velocity,  or  tend  to  move  with  the  same  velocity,  it  is  evi- 
dent, I  say,  that  the  resultant  of  all  these  motions  or  tendencies 
would  pass  through  the  centre  of  gravity  of  the  body  or  system, 
and  that  consequently  the  system  would  move,  or  tend  to  move, 
as  if  the  several  masses  were  all  concentrated  at  the  centre  of 
gravity ;  and  were  together  urged  with  a  velocity  equal  to  that  which 
urges  each  of  the  parts. 

1 1 G.  We  must  infer  reciprocally,  that  if  any  force  be  applied 
at  the  centre  of  gravity  of  a  system  of  bodies ;  all  the  equal  parts 
of   this  system   will    partake    equally  of  this  motion,  and  will  all 
proceed  with  an  equal  velocity,  obtained  by   dividing  the   quantity      gs 
of  motion  applied  at  this  centre  by  the  entire  mass  of  the  system, 


70  Statics. 

and  this  velocity  will  have  for  its  direction  that  of  the  force  applied 
at  the  centre  of  gravity. 

Indeed,  whatever  be  the  motions  distributed  among  the  parts 
of  the  system,  we  see  clearly  tliat  they  must  have  for  a  resultant 
the  very  force  applied  at  the  centre  of  gravity,  since  it  is  supposed 
that  the  system  is  free,  and  that  there  is  consequently  nothing  to 
destroy  any  part  of  the  force  thus  applied. 

117.  Also,  since  several  forces  applied  at  the  same  point,  re- 
duce themselves,  by  the  preceding  principles,  to  a  single  one,  we 
infer  generally,  that  whatever  be  the  number,  direction,  and  magni- 
tude of  the  forces  which  are  applied  at  the  centre  of  gravity  of  a 
system  of  bodies  ; 

(1).  All  parts  of  this  system  will  have  the  same  velocity  ; 

(2).  This  velocity  will  be  in  the  direction  of  the  resultant  of  all 
the  applied  forces  ; 

(3).  It  will  be  equal  to  the  quantity  of  motion,  which  this  resul- 
tant represents,  divided  by  the  entire  mass  of  the  system. 

1 18.  Whence  we  conclude,  that  while  the  forces  which  act  upon 
a  body,  are  capable  of  bein^  reduced  to  a  single  one,  the  direction  of 
which  passes  through  the  centre  of  gravity,  this  body  will  not  turn 
about  the  centre  of  gravity. 

119.  But  if  the  forces  which  act  upon  a  body  cannot  be  re- 
duced to  a  single  one,  or  on  the  supposition  that  they  admit  of 
being  so  reduced,  if  the  direction  does  not  pass  through  the  cen- 
tre of  gravity,  all  the  parts  of  the  system  will  not  have  a  common 
motion.  Nevertheless  the  centre  of  gravity  will  move  in  the  same 
manner  as  if  all  the  forces  were  applied  directly  at  this  point,  as  we 
now  propose  to  show. 

Fig.  53.  120.  Let  us  in  the  first  place  suppose  three  bodies  m,  n,  o, 
moving  in  parallel  lines  AA",  BB",  CC",  (situated  in  the  same  or 
in  different  planes,)  and  with  velocities  represented  by  the  lines 
AA",  BB",  CC",  respectively,  the  motion  of  each  being  uniform. 
Let  us  suppose  also,  that  G  is  the  centre  of  gravity  of  these  bodies, 
when  they  are  in  A,  B,  C ;  and  that  G"  is  their  centre  of  gravity, 
when  they  are  in  A",  B",  C",  where  they  will  arrive  in  the  same 
time,  since  their  velocities  are  represented  by  AA",  BB",  CC" ; 


Properties  of  the  Centre  of  Gravity.  71 

joining  GG"^  I  say  that  this  line  will  be  parallel  to  AA",  BB",  Sic., 
and  that  it  will  represent  the  course  described  by  the  centre  of 
gravity  during  the  supposed  motion  of  the  bodies  m,  n,  o,  and  that 
it  will  be  described  uniformly. 

(1).  It  is  evident  that  the  course  described  by  the  centre  of 
gravity  will  be  parallel  to  the  lines  AA",  BB",  &;c. ;  for  at  what- 
ever point  we  suppose  it  at  any  instant,  if  we  imagine  a  plane  pass- 
ing through  it,  the  sum  of  the  moments  with  respect  to  this  plane 
must  be  zero.  Now  if  we  conceive  a  plane  parallel  to  the  direc-  85. 
lions  of  the  bodies  m,  n,  a,  and  passing  through  G,  the  moments 
with  respect  to  this  plane  cannot  but  be  zero  during  the  whole 
motion,  for  the  bodies  in  their  motion  are  supposed  not  to  alter 
their  distances  from  this  plane  ;  their  distances  are  therefore  con- 
stantly the  same,  and  consequently  these  moments  are  also  con- 
stantly the  same ;  but  at  the  commencement  of  the  motion,  that  is, 
when  the  centre  of  gravity  is  in  G,  the  sum  of  the  moments  is 
.zero ;  accordingly,  it  is  still  zero  in  whatever  part  of  their  directions 
the  bodies  are  ;  the  centre  of  gravity  is  consequently  in  a  plane 
parallel  to  the  directions  of  the  bodies  and  passing  through  the  first 
situation  G  of  this  centre.  And,  as  in  the  reasoning  here  usedy 
the  position  of  this  plane  is  not  otherwise  determinate  than  that  it 
must  be  parallel  to  the  directions  of  the  bodies  m,  w,  o,  and  pass 
through  the  point  G ;  it  may  be  shown,  in  like  manner,  that  this 
centre  is  in  any  other  plane  parallel  to  the  directions  of  the  bodies 
and  passing  through  the  point  G  ;  it  is  consequently  in  the  common 
intersection  of  these  planes  ;  therefore  the  centre  of  gravity  moves 
according  to  GG'%  parallel  to  the  directions  of  the  supposed  bodies. 

(2).  The  centre  of  gravity  moves  uniformly  ;  that  is^  if  when 
the  bodies  m,  n,  o,  he,  have  arrived  at  A',  B',  O,  he,  we  sup- 
pose that  the  centre  of  gravity  is  in  G',  we  shall  have 

GG"  :   GG'   :  :  AA"   :   AA'   :  :  BB"   :  BB'  :  :   &c. ; 
in  other  words,  the  spaces  described  in  the  same  time  by  the  centre 
of  gravity  and  the  several  given  bodies  will  be  as  their  velocities 
respectively. 

Indeed,  if  we  conceive  a  plane  represented  by  ZIL  to  which  the 
directions  of  the  several  motions  are  perpendicular ;  we  shall  have, 
by  the  nature  of  the  centre  of  gravity,  76. 

m  X  HA  -f  n  xIB-\-oxLC={m-\-n  +  o)xKGj 


72  Statics. 

and  for  the  same  reason,  when  they  are  at  A'^,  B",  C", 
m  X  HA"  +  71  X  IB"  -\-o  X  LC"  =  (m  +  n  +  o)  X  KG''. 
U  from  the  second  of  these  equations,  we  subtract  the   first,   bear- 
ing in  mind  that  HA"  —  HA  =  AA",    IB"  —  IB  =  BB",  &c., 
we  shall  have 

OT  X  AA"  -\-7iX  BB"  —  ox  CC"  —  (ni  +  n  -\-  o)  X  GG"  ; 
and  for  the  same  reason,  when  they  are  at  A',  B',  O, 
m  X  AA'  +  n  X  BB'  —  o  X  CC  =  (m  +  «  +  o)  X  GG'. 

Now  since  AA',  BB',   CO,   are   described   uniformly  in  the 
26.     same  time,  these  spaces  must  be  as  the  velocities,  AA",  BB",  CC"  ; 
consequently 

AA"  :  BB"  : :  AA'  :  BB',      AA"  :  CC"  : :  AA'  :  CO, 

II-      RR/       AAxjm'    ^^,        AA'  X  CO" 

which  give  BB'  = -j-^, ,  CO  =  --^, . 

Substituting  these  values  in  the  last  of  the  above  equations,  we  shall 

have 

^  ^,  ,          AA'xBB"             AA' X  CC"      ,     ,      ,    ,     ^>^, 
mX'^''I'+  n  X ^j. ox  — -£^, ={m-\-n-\-o)  X  GG' 

or,  by  making  the  denominator  to  disappear, 

[m  X  A  A"  -\-  n  X  BB"  —  oX  CC")  X  AA' 
—  {m  -{-  n  +  o)  X  GG'  X  AA". 
This  equation  divided  by  that  in  which  GG"  enters,  gives 

AA'  =         ii,    — ,  or  A  A'  X  GG"  =  GG'  X  AA", 

from  which  we  have 

GG"  :   GG'   ::   AA"   :  AA', 
which  was  proposed  to  be  demonstrated. 

We  remark  that  the  equation  in  which  GG"  enters,  gives 

^  ^„  _  m  X  AA"  +  »  X  BB"  —  0  X  CC" 

m  -\-  n  -j-  0 

Now  the  lines  AA",  BB",  CC",  GG",  are  the  velocities  re- 
spectively of  the  bodies  m,  n,  o,  and  of  the  centre  of  gravity  G ; 
consequendy  m  X  AA",  n  X  BB",  6lc.,  are  the  quantities  of 
motion  respectively.  Accordingly,  since  the  reasoning  we  have 
pursued  does  not  depend  in  any  degree  upon  the  number  of  bodies, 
we  infer,  as  a  general  conclusion, 


Properties  of  the  Centre  of  Gravity.  73 

(1.)  That,  if  any  number  of  bodies  describe  parallel  lines,  the 
centre  of  gravity  describes  a  line  parallel  to  them  ; 

(2.)  That  the  velocity  of  the  centre  of  gravity  is  equal  to  the 
sum  of  the  quantities  of  motion  of  the  bodies  moving  in  one  direction, 
minus  the  sum  of  the  quantities  of  motion  of  those  that  move  in  the 
opposite  direction,  divided  by  the  sum  of  the  masses. 

121.  If  any  one  of  the  bodies  be  at  rest,  the  velocity  of  this 
body  will  be  zero,  and  the  quantity  of  motion  also  will  be  zero. 
Thus  it  will  disappear  from  the  numerator  of  the  fraction  which  ex- 
presses the  velocity  of  the  centre  of  gravity ;  but  the  denominator, 
remaining  unchanged,  will  in  every  case  be  the  sum  of  all  the 
masses. 

122.  If  the  sum  of  the  quantities  of  motion  of  the  bodies  which 
move  in  one  direction,  be  equal  to  the  sum  of  the  quantities  of 
motion  of  those  moving  in  the  opposite  direction,  the  nunierator  of 
the  fraction  which  expresses  the  velocity  of  the  centre  of  gravity 
will  be  zero.  This  centre  of  gravity,  therefore,  will  be  at  rest. 
Accordingly,  whatever  be  the  parallel  motions  of  several  bodies, 
their  common  centre  of  gravity  will  remain  at  rest,  when  the  sum  of 
the  quantities  of  motion  of  those  that  move  in  one  direction  is  equal 
to  the  sum  of  the  quantities  of  motion  of  those  that  move  in  the 
opposite  direction. 

123.  Since  the  quantities  of  motion  represent   the  forces  ;  and     28. 
the  resultant  of  any  number  of  parallel  forces  is  equal   to  the   sum 

of  those  which  act,  or  tend  to  act,  in  one  direction,  minus  the  sum 
of  those  which  act,  or  tend  to  act,  in  the  opposite  direction  ;  we  50, 
conclude,  that,  if  any  number  of  parallel  forces  are  applied  to 
different  parts  of  a  system  of  bodies,  the  centre  of  gravity  of  this 
system  will  move  as  if  the  forces  in  question  ivere  all  applied  directly 
at  this  point. 

124.  Let  there  be  any  number  of  bodies  moving  according  to 
any   given   straight    lines.      If   we    imagine    three  rectangular  co- 
ordinates, we  may  always  decompose  the  velocity  of  each  body  into 
three  other    velocities,   parallel    respectively  to   these   three    lines.     73. 
Now  it  follows  from  what  we  have  just  said,  that  the   motion   of 

the  centre  of  gravity,  in  virtue  of  the  motions  parallel  to  one  of 
Mech:  10 


74  Statics. 

these  lines,  will  be  parallel  to  this  same  line  ;  it  will  also  be  uniform, 
and  equal  to  the  sum  of  the  quantities  of  motion  (estimated  parallel 
to  this  line),  divided  by  the  sum  of  the  masses.  If  therefore  we 
suppose  that  the  motion  of  the  centre  of  gravity,  parallel  to  each  of 
these  lines,  is  thus  determined,  and  that  these  three  motions  are 
afterwards  reduced  to  one  (which  may  be  done,  since  they  are  all 
72.  applied  at  the  same  point),  we  shall  have  the  course  of  the  centre 
of  gravity  in  a  single  line.  Also,  as  the  elements  here  employed, 
are  simply  the  forces  themselves  which  the  bodies  have  parallel  to 
the  three  co-ordinates,  and  as  the  single  force  of  the  centre  of 
gravity  is  thus  found  to  be  composed  of  the  resultant  forces  parallel 
respectively  to  these  given  lines,  it  cannot  but  be  equal  and  parallel 
to  the  resultant  of  all  the  forces  applied  to  the  bodies  in  question  ; 
hence,  whatever  be  the  directions  and  magnitudes  of  the  forces  ap- 
plied to  different  parts  of  a  system  of  bodies,  the  centre  of  gravity 
moves  always,  or  tends  to  move,  in  the  same  manner,  as  if  the  forces 
in  question  luere  all  applied  directly  at  this  point. 

125.  In  the  foregoing  article,  we  have  said  that  we  may  ahvays 
decompose  the  velocity  of  each  body  into  three  others,  parallel 
respectively  to  three  lines  whose  position  is  given.  If  the  direction 
of  one  of  the  bodies,  however,  be  parallel  to  the  plane  of  two  of 
the  three  assumed  lines,  or  if  it  be  parallel  to  one  of  these  lines,  it 
might  seem  that,  in  the  first  case,  it  would  not  admit  of  being  de- 
composed, except  into  two  forces,  parallel  to  two  of  the  three  given 
lines  ;  and  that  in  the  second  case,  no  decomposition  whatever  could 
take  place  into  forces  parallel  to  the  two  other  lines.  Notwithstand- 
ing this  apparent  difficulty,  the  proposition  is  true  universally.  We 
Fig.  54.  see,  for  example,  that  so  long  as  the  line  AB  is  not  parallel  to 
either  of  the  lines  XZ,  XT,  we  can  always  decompose  the  force 
represented  by  AB  into  two  others,  AC,  AD,  parallel  to  these  two 
lines  respectively ;  but  we  perceive,  at  the  same  time,  that  the  more 
AB  approaches  to  a  parallelism  with  XT,  the  more  the  force  AD 
diminishes  ;  so  that  it  becomes  zero,  when  AB  is  parallel  to  XT. 
There  is  not,  therefore,  in  this  case,  the  less  propriety  in  supposing 
a  decomposition  into  two  forces,  because  one  of  them  is  zero.  For 
a  like  reason,  we  may,  in  the  same  case,  suppose  a  decomposition 
into  three  forces,  parallel  to  three  given  lines  XT,  XZ,  XY,  two  of 
which  are  equal  respectively  to  zero. 


General  Principle  of  the  Equilibrium  of  Bodies.  75 

126.  From  what  we  have  now  said,  taken  in  connexion  with 
that  of  article  122,  we  infer,  that  the  centre  of  gravity  of  a  system 
of  bodies  will  remain  at  rest,  if  each  of  the  forces  applied  to  the 
several  parts  being  decomposed  into  three  other  forces  parallel  re- 
spectively to  three  rectangular  co-ordinates,  the  sum  of  the  forces,  or 
quantities  of  motion,  parallel  to  each  of  these  three  lines  he  equal  to 
zero,  the  forces  which  act  in  opposite  directions  being  taken  with 
contrary  signs. 

127.  When  all  the  forces  are  in  the  same  plane,  it  is  evidently 
sufficient  to  decompose  each  force  into  two  others  parallel  to  two 
assumed  lines,  these  lines  being  perpendicular  to  each  other,  and 
drawn  in  the  same  plane  with  the  given  forces;  for  the  forces  which 
are  perpendicular  to  this  plane  being  zero,  the  motion  of  the  centre 
of  gravity  in  virtue  of  these  forces  is  also  zero. 

128.  In  all  that  we  have  said,  we  have  supposed  each  of  the 
bodies  which  compose  the  system,  to  obey  fully  and  freely  the  force 
by  which  it  is  urged.  But  the  same  principles  hold  true  no  less 
when  the  bodies  are  constrained  in  their  motions,  provided  the  ob- 
stacles do  not  proceed  from  a  force  foreign  to  the  system,  that  is, 
provided  there  are  no  impediments  except  those  which  arise  from 
the  difficulty  of  yielding  to  these  motions  by  the  manner  in  which 
they  are  disposed  among  themselves  or  connected  with  each  other. 
This  we  propose  to  demonstrate  after  having  first  made  known  the 
general  law  of  the  equilibrium  of  bodies  and  the  general  law  of  their 
motion. 


General  Principle  of  the  Equilibrium  of  Bodies. 

129.  Whatever  be  the  forces  [acting  or  resisting),  applied  to  a 
body,  to  a  system  of  bodies,  to  a  machine,  ^c,  and  whatever  be  the 
directions  of  these  forces,  if  we  conceive  that  each  is  decomposed 
into  three  others  parallel  respectively  to  three  rectangular  co- 
ordinates, it  is  necessary  in  order  that  all  these  forces  should  be  in 
equilibrium,  that  the  sum*  of  the  forces  which  act  parallel  to  each 
of  these  co-ordinntes,  should  be  equal  to  zero. 


*  By  sum  of  the  forces  in  what  follows,  is  to  be  understood 
the  sum  of  those  which  act  in  one  direction,  minus  the  sum  of 
those  which  act  in  the  opposite  direction. 


76  Statics* 

Indeed,  whatever  be  the  number  and  nature  of  the  forces,  we 
have  seen  that  they  may  ahvays  be  reduced  to  three,  the  directions 
73.      of  which  are  parallel  to  three  rectangular   co-ordinates.     If  there- 
fore we  suppose  an  equilibrium  among  all  the  forces  of  the  system, 
it  is  necessary  that  there  should   be  an   equilibrium   among  these 
three  resultants,  or  that  each  resultant  should   be  equal  to   zero. 
Now  these  resultants,  being  perpendicular  to  each  other,  can  neither 
35.      increase  nor  diminish  one  another.     But  each  is  equal  to  the  sum  of 
70.      the  partial  forces  parallel  to  it;  therefore,  there  being  an  equilibrium, 
the  sums  of  the  forces  which  by  decomposition  are  found  to  act  in 
a  direction   parallel  respectively  to  three  rectangular  co-ordinates 
must  each  be  equal  to  zero. 

130.  If  all  the  forces  are  exerted  in  the  same  plane,  the  sum 
of  each  of  the  forces  which  by  decomposition  are  found  to  be 
parallel  respectively  to  two  co-ordinates,  drawn  in  this  plane,  will  be 
zero.  Moreover,  if  all  the  given  forces  should  happen  to  be 
parallel  to  each  other,  the  sum  of  their  forces  must  be  zero. 
These  two  cases  are  evidently  comprehended  in  the  general  propo- 
sition. 

131.  It  should  be  remarked,  that  this  proposition  holds  true, 
whatever  be  the  case  in  which  the  equilibrium  occurs ;  but  we 
should  err  by  supposing  that  it  is  sufficient  in  order  that  an  equilibri- 
um may  take  place.  The  other  conditions  necessary  for  this  effect 
vary  according  to  the  particular  qualities  or  disposition  of  the  parts 
of  the  system  or  machine  in  question. 

132.  The  proposition,  moreover,  holds  true,  whether  the  forces 
which  are  applied  to  the  different  parts  of  the  system  are  all  active, 
or  whether  some  are  active,  and  others  merely  capable  of  resisting, 
as  supports,  fixed  points,  surfaces,  &.C.,  which  oppose  the  action  of 
forces ;  for  impediments  by  destroying  motion  are  equivalent  in  this 
respect  to  active  forces. 

jyAlemheri's  Principle^  and  concluding  Deductions. 

133.  Whatever  he  the  manner  in  which  several  bodies  come  to 
change  their  existing  state  as  to  motion,  if  we  conceive  the  motion 
which  each  body  would  have  the  following  instant,  on  the  supposition 


D^Alemberfs  Principle.  77 

of  its  being  free,  as  decomposed  into  two  others,  one  of  which  is  that 
which  the  body  actually  has  after  the  change,  the  second  must  be  such, 
that  if  each  of  the  several  bodies  had  had  no  other  than  this,  they 
would  have  remained  in  equilibrium. 

This  proposition  must  be  admitted,  since,  if  the  second  motions 
be  not  such  that  an  equilibrium  would  result  from  them  in  the 
system,  the  first  component  motions  cannot  be  those  that  the  bodies 
are  considered  as  having  after  the  change,  for  these  would  neces- 
sarily be  altered  by  such  a  supposition. 

134.  Let  us  suppose  now  that  several  bodies,  either  free  or 
connected  together  in  any  manner  whatever,  come  to  receive  certain 
impulses  which  they  cannot  entirely  obey  on  account  of  a  reciprocal 
restraint,  the  centre  of  gravity  will  move  as  if  all  the  bodies  were 
free. 

Indeed,  whatever  be  the  motion  which  each  part  of  the  system 
has,  we  may  always  conceive  tliat  which  is  impressed  upon  it  as 
composed  of  two  parts,  namely,  that  which  it  actually  takes,  and  a 
second.  But  in  virtue  of  these  second  motions,  the  system  must  be  40. 
in  equilibrium  ;  if  we  suppose,  therefore,  these  second  motions  dc' 
composed  each  into  three  others,  parallel  to  three  rectangular  co 
ordinates,  the  sum  of  the  forces  which  would  result  from  this 
decomposition,  parallel  to  each  of  the  three  co-ordinates,  must  be 
zero.  Now  the  course  which  the  centre  of  gravity  tends  to  describe  129 
in  virtue  of  each  of  these  forces,  is  equal  to  the  sum  of  the  forces 
parallel  respectively  to  each  of  the  co-ordinates,  divided  by  the 
sum  of  the  bodies.  Consequently  the  course  which  it  tends  120, 
to  describe  in  virtue  of  the  changes  arising  in  the  system 
from  the  reciprocal  action  of  the  {)arts  is  zero  ;  accordingly 
the  centre  of  gravity  does  not  partake  of  these  changes,  that 
is,  it  moves  as  if  each  of  the  several  parts  of  the  system 
obeyed  freely,  and  without  loss,  the  force  by  which  it  is  urged. 
Therefore,  the  state  of  the  centre  of  gravity  of  a  body,  or 
system  of  bodies,  does  not  change  by  the  reciprocal  action  of  the 
parts  of  this  body  or  system. 

135.  Hence  we  infer  ;  (1.)  That,  if  a  body  or  system  of  bodies 
turn  about  its  centre  of  gravity  in  any  manner  whatever,  this  centre 


133. 


78  Statics. 

of  gravity  will  remain  continually  in  the  same  state,  as  if  the  body 
did  not  turn. 

Moreover,  from  this  same  principle  and  that  of  article  124  we 
conclude,  that 

136.  (2.)  If  any  body,  whatever  be  its  figure,  or  any  assem- 
blage of  bodies,  receive  an  impulse  in   any   direction  whatever  as 

Fig.  55.  ^B,  which  transmits  itself  entirely  to  the  body  ;  the  centre  of  grav- 
ity G  will  move  according  to  a  line  TS  parallel  to  AB,  in  the  same 
manner,  as  if  this  force  were  immediately  applied  at  the  centre  of 
gravity  in  this  direction.  And  if  several  forces  act  at  the  same 
time  upon  different  points  of  this  body,  the  centre  of  gravity  will 
move  as  if  all  the  forces  in  question  were  applied  directly  at  this 
point. 

137.  If,  therefore,  at  the  instant  the  body  receives  an  impulse  in 
the  direction  AB,  we  apply  at  the  centre  of  gravity  in  the  op- 
posite direction  SG,  a  force  equal  to  that  which  acts  according  to 
AB,  the  centre  of  gravity  will  remain  at  rest.  Nevertheless  it  is 
evident  that  the  other  parts  of  the  body  v/ould  not  remain  at  rest, 
since  these  two  forces,  although  equal,  are  not  directly  opposite  to 
each  other.  Now  the  only  motion  which  this  body  can  have,  its 
centre  of  gravity  remaining  at  rest,  is  evidently  a  motion  of  rotation 
about  its  centre  of  gravity. 

Therefore,  if  a  body  receive  one  or  several  impulses,  in  direc- 
tions which  do  not  pass  through  the  centre  of  gravity ;  (1.)  This 
centre  of  gravity  will  move  as  if  all  the  forces  were  applied  directly 
at  this  point,  each  in  a  direction  parallel  to  that  which  it  actually 
has.  (2.)  The  parts  of  this  body  will  turn  about  their  centre  of 
gravity,  as  they  would  do  by  virtue  of  the  forces  which  are  actually 
applied  to  the  body,  if  the  centre  of  gravity  were  fixed. 

138.  We  infer,  moreover,  that  if  the  state  of  the  centre  of  gravity 
of  a  body  undergoes  a  change,  this  can  proceed  only  from  the  ac- 
tion or  resistance  of  new  forces  foreign  to  this  body  ;  and  that  con- 
sequently this  change  is  always  determined  by  seeking  the  resultant 
which  all  the  forces  would  have,  if  they  were  applied  to  the  centre 
of  gravity,  each  in  a  direction  parallel  to  that  which  it  actually  has. 


Rope  Machine.  79 


1^ 


Application  of  the  Principles  of  Equilibrium  to  the  Machines  usual- 
ly denominated  Mechanical  Powers. 

139.  The  general  object  of  machines  is  to  transmit  the  action  of 
forces.  The  end  to  be  attained  is  not  always  to  augment  the  action 
of  which  the  power  employed  is  capable,  when  applied  directly  to 
the  mass  to  be  moved,  or  resistance  to  be  overcome.  Sometimes 
it  is  merely  proposed  to  transmit  this  action  in  a  determinate  direc- 
tion. At  other  times  the  purpose  to  be  answered  is  to  cause  a  body 
to  describe  spaces  regulated  upon  certain  conditions,  relative  either 
to  time  or  other  circumstances,  conditions  wl)ich  do  not  always 
require  that  the  force  employed  should  augment  as  it  is  transmitted^ 
We  have  examples  of  this  kind  of  machinery  in  cbcks,  watches, 
orreries,  &c. 

The  number  and  nature  of  the  machines  vary  according  to  the 
object  we  have  in  view.  But  to  be  able  to  determine  their  effects,, 
it  is  not  necessary  to  consider  them  all  separately.  However  com- 
pounded and  varied  they  may  be,  they  are  merely  combinations  of 
a  certain  very  limited  number  of  simple  machines.  We  come  now 
to  make  known  the  properties  of  these  simple  machines.  We  shall 
afterward  proceed  to  show  how  these  properties  are  to  be  applied  in 
estimating  the  effects  of  compound  machines. 

There  are  now  usually  reckoned  seven  simple  machines,  namely, 
the  rope  machine,  the  lever,  the  pulley,  the  wheel  and  axle,  the  in- 
clined plane,  the  screw,  and  the  wedge. 

These  machines,  being  considered  simply  with  respect  to  a  state 
of  equilibrium,  may  be  reduced  to  two,  and  indeed  to  one,  namely, 
the  lever.  But  in  the  case  of  motion,  the  nature  of  each  leads  to 
particular  considerations,  and  requires  a  separate  treatment. 


Of  the  Rope  Machine. 

140.  We  proceed  on  the  supposition  that  the  ropes  or  cords 
employed  are  perfectly  flexible.     It  will  be  shown,  however,  in  the 


so  Statics. 

sequel,  what  allowance  is  to  be  made  for  the  want  of  this  quality. 
Moreover,  cords  are  first  considered  as  destitute  of  weight,  regard 
being  had  afterward  to  their  gravity.  The  greater  or  less  diameter 
of  the  cords  also  is  not  considered  as  affecting  the  communication 
of  forces ;  since  we  may  always  substitute  in  imagination  for  these 
cords,  considered  as  cylinders,  a  line  or  thread  answering  to  their 
axes,  the  force  employed  being  considered  as  acting  by  means  of 
this  thread  only. 

We  employ  cords  to  transmit  the  action  of  forces  immediately, 
or  in  connexion  with  machines.  But  in  order  to  judge  of  the  effects 
of  powers  applied  to  machines  by  means  of  cords,  it  is  necessary  to 
ascertain  the  effects  of  which  these  powers  are  capable,  when  they 
act  by  means  of  cords  alone. 

Fig. 56.  141.  Accordingly,  let  us  consider  three  powers,/?,  q,  r,  as  act- 
ing one  against  another,  by  means  of  three  cords  A  p,  A  g,  A  r, 
united  at  A  by  a  knot;  and  supposing  the  directions  A p,  A  q, 
A  r,  to  be  known,  we  propose  to  determine  the  conditions  necessa- 
ry to  an  equilibrium  among  these  forces,  and  the  ratio  of  these 
forces. 

(1.)  It  is  evident,  in  the  first  place,  that  they  must  all  three  be 
in  the  same  plane  ;  for  if  one,  the  force  r,  for  example,  were  not  in 
the  plane  of  the  two  others,  we  could  always  conceive  it  decom- 
posed into  two  forces,  one  in  this  plane,  and  the  other  jwpendicular 
to  this  plane,  and  consequently  perpendicular  to  eac^of  the  two 
40.  forces  p,  q ;  this  perpendicular  force  would  not  act,  therefore,  in 
any  way  against  the  forces  p,  q ;  and  would  accordingly  have 
nothing  to  be  opposed  to  it,  and  an  equilibrium  with  respect  to  it 
could  not  take  place. 

(2.)  These  three  forces  being  then  in  the  same  plane,  it  is 
necessary,  in  order  that  they  may  be  in  equilibrium,  that  some  one 
of  them,  the  force  p  for  example,  should  produce  two  efforts,  the 
one  equal  and  opposite  to  the  force  q,  and  the  other  equal  and  op- 
posite to  the  force  r.  Now  if,  after  having  produced  i-A,  qA,  we 
take  any  line  AD  to  represent  the  force  ^,  and  upon  AD  as  a  diag- 
onal we  construct  the  parallelogram  ACDB,  the  tv;o  sides  AB,  AC, 
will  represent  two  forces,  which  acting  conjointly  according  to  these 
40.  directions,  would  produce  the  same  effect  as  the  force  p.  Ac- 
cordingly, AB,  AC,  are  the  efforts  that  p  actually  opposes  to  the 


Rope  Machine.  81 

two  forces  q  and  r ;  hence,  in  order  that  there  may  be  an  equi- 
librium, it  is  necessary  that  q  should  be  represented  by  BA  and  r 
by  CA,  p  by  supposition  being  represented  by  AD.  We  have, 
therefore,  the  following  propoi'tions, 

p  :  q  ::  AD  :  AB,     and    p  :  r    :  :    AD  :  AC; 

that  is, 

p  :  q  :  r  :  :  AD  :  AB  :  AC. 

Such  is  the  ratio  that  must  exist  among  the  forces  p,  q,  r,  In 
order  that  an  equilibrium  may  take  place. 

142.  Since  the  two  forces  q,  r,  must  be  equal  to  the  two  forces 
AB,  AC,  which  are  the  components  of  the  force  p,  we  infer  that, 
when  tliere  is  an  equilibrium  among  three  forces,  any  two  of  them 
must  have  the  same  ratio  to  the  third,  that  two  components  have  to 
their  resultant. 

143.  Accordingly  we  have  the  proportion,  ^• 
p  :  q  :  r  ::  sin  BAC  :  sin    CAD  :  sin  DAB 

:  :  sin  q  A  r   :  sin    r  AS  :  sin  q  AS, 
p  A  being  produced  toward  S  ;  that  is,  ivhen  three  forces  are  in,  equi- 
librium, each  is  represented  by  the  sine  of  the  angle  comprehended 
between  the  directions  of  the  two  others  ;  these  directions  being  pro- 
duced if  necessary. 

144.  Since  the  three  forces  p,  q,  r,  which  are  to  be  in  equilibri- 
um, arc  represented  by  AD,  AB,  AC,  or,  which  amounts  to  the 
same  thing,  by  the  sides  AD,  AB,  BD,  of  the  triangle  ABD,  of 
which  the  angles  ABD,  BDA,  DAB,  are  equal  to  the  angles  CA  q, 
r  AS,  q  AS,  determined  by  the  directions  of  the  forces,  it  will  be 
seen  that  all  the  questions  which  can  occur  with  respect  to  the 
value  and  direction  of  the  forces,  requisite  to  an  equilibrium,  refer 
themselves  to  the  subject  of  trigonometry.  If,  for  example,  the 
values  of  tiirec  forces  p,  q,  r,  were  given,   and  it  were  proposed 

to  find  their  direction,  we  should  resolve  the  triangle  DBA,  the  Trig.  88. 
three  sides  of  wiiich  would  be  known,  and  the  angles  thus  obtained 
would  give  the  directions  of  the  forces  required.  If  we  had  given 
the  two  forces  p,  q,  and  the  angle  p  A  q,  o(  their  directions,  or  its 
supplement  q  AS  =^  DAB  ;  then  we  should  have  the  two  sides  AB 
AD,  and  the  contained  angle  DAB,  from  which  we  should  readily 
Mech.  J I 


82  Statics. 

determine  the  side  DB,  or  the  force  r,  and  the  angle  BDA,  or  its 
equal  r  AS,  formed  by  the  directions  of  r  and  p.  If  the  angles 
Trig.  35.  formed  by  the  directions  of  the  three  forces  were  given,  we  could 
not  thence  determine  the  absolute  values  of  the  three  forces,  but 
only  their  ratio  to  each  other.  In  all  other  cases,  the  proposition 
143.  above  established  will  be  sufficient  for  a  complete  solution,  wiien 
three  things  only  are  given. 

145.  If  instead  of  having  two  forces,  q  and  r,  attached  to  two 
cords,  these  two  cords  were  firmly  fixed  at  q  and  r,  or  at  any  points 
respectively  in  their  directions,  JIB,  AC,  would  express  the  efforts 
supported  by  these  points. 

Fig.  56.  140.  We  have  supposed  the  three  cords  firmly  attached  by  a 
Fig.  o7.  \^^Q[  ji  j3j^  jc  {[jg  (.Q(.(j  ff)  which  the  power  p  is  applied  had  a 
ring  at  its  extremity  A,  through  which  the  cord  q  A  r  passed,  we 
should  not  be  able  to  assign  the  directions  of  the  three  cords. 
Indeed  it  is  not  sufficient,  in  this  case,  that  the  effort  AB  has  the 
direction  q  A,  and  is  equal  to  the  force  q,  andt  hat  ^Chas  the  di- 
rection r  A,  and  is  equal  to  r ;  it  is  necessary,  further,  that  the  ring 
should  not  slip  upon  the  cord  q  A  r,  which  requires  that  the  angle 
q  AS  should  be  equal  to  SA  r ;  that  is,  that  the  power  p  should  be 
directed  in  such  a  manner,  as  to  bisect  the  angle  q  A  r.  But  we 
have  always 

p  :  q  :  r  '.:  ^m  q  A  r  :  sin  r  AS  :  sin  q  AS  j 

and  as  r  AS  =.  q  AS  =l  \  q  A  r,  this  series  of  ratios  becomes 
p  :  q  :  r  ::  z\n  q  A  r  :   sin  \  q  A  r  :  %m  \  q  A  r  ', 

so  that  the  two  powers  q  and  r  are  equal. 

147.  The  same  result  would  follow,  if  the  cord  q  A  r,  drawn  by 
the  two  powers  r,  q,  passed  over  a  fixed  point  A.  The  two  powers 
r,  q,  must  be  equal,  and  the  force  exerted  by  them  upon  the  fixed 
point  will  be  directed  in  such  a  manner  as  to  bisect  the  angle  q  A  r, 
and  its  magnitude  will  be  with  respect  to  each  of  these  two  powers, 
as  the  sine  o{q  A  r  is  to  the  sine  of  half  q  A  r, 

148.  The  foregoing  articles  being  well  understood,  it  will  be 
easy  to  determine  the  conditions  of  equilibrium  among  as  many 
powers  as  we  choose  to  employ,  applied  to  different  cords,  and 
united  by  the  same  or  by  different  knots. 


Rope  Machine.  83 

Let  us  suppose,  in  the  first  place,  that  each  knot  connects  only 
three  cords,  and  that  they  are  all  in  the  same  plane,  as  represented 
in  figure  58. 

The  power  p  is  exerted  against  the  two  cords  A  q,  AB.  Let 
the  directions  of  these  cords  be  produced  ;  having  taken  AF  to 
represent  the  power  j).  we  form  upon  AF  as  a  diagonal,  and  upon 
the  prolongations  AE,  AD,  as  sides,  the  parallelogram  ADFE. 
The  force  g  will  be  expressed  by  AE,  and  the  tension  of  the  cord 
BA  by  AD ;  so  that,  denoting  by  a  this  tension,  we  shall  have 

p  :  Q  :  a  ::AF  :     AE         :  AD 

sin  DAE  :  sin  FAD  :  sin  FAE 
sin  g  AD  :  sin  FAD  :  sin  FAE. 

Suppose  the  effort  AD  applied  at  B,  according  to  BI,  in  the 
same  straight  line  with  AD,  and  equal  to  AD.  The  force  BI\s 
exerted  against  the  power  q,  and  against  the  cord  BC.  By  pro- 
ducing, therefore,  as  above,  the  cords  q  B,  CB,  and  forming  the 
parallelogram  GBHI,  BIl  will  represent  the  value  of  the  force  q, 
and  Z?G  the  tension  of  the  cord  CB.  We  shall  accordingly  have, 
b  denoting  this  tension, 

a  :  q  :  h  ::  sm  GBH,         sin  IBG  :  sin  IBH. 

Suppose  the  e^on  BG  applied  at  C,  according  to  CK,  in  a 
straight  line  with  BG,  and  equal  to  BG.  The  force  CK  is  exert- 
ed against  s?  and  against  r.  If  therefore  we  produce  r  C,  vs  C,  and 
form  as  before  the  parallelogram  c/17CLA',  CM  will  express  the  value 
that  must  belong  to  the  force  ;•,  and  CD  that  which  must  be  exerted 
by  CT  ;  whence, 

6  :  r  :  nr  : :  sin  LCM  :  sin  KCL  :  sin  MCK. 

If  we  would  have  immediately  the  ratio  of  the  tension  q  of  any 
branch  q  A  ol  the  cord  to  the  tension  of  any  other  branch,  C  vs,  for 
example,  it  may  be  readily  obtained  in  the  following  manner. 

Of  the  series  of  ratios  above  found,  if  we  take  only  those  which 
relate  to  the  tensions  of  the  parts  of  the  cord,  g  ABC  to,  we  shall 
have 

g  :  a  ::  s\n  FAD  :  sin  FAE, 
a  :  b  ::sm  GBH  :  sin  IBJJ, 
b   :  cT  :  :  sin  ECM  :  sin  MCh  ; 


p 

:  a   : 

:  sin 

qJD 

a   : 

:  q   : 

:  sin 

GBH 

'i  ' 

:  6   : 

:  sin 

IBG 

h 

:  r  : 

:  sin 

LCM 

84  Statics. 

these  being  multiplied  in  order,  we  have 

p  :  C7  : :  sin  FAD  sin  GBH s'm  LCM :  sin  FAE  sin  IBH  sin  MCK. 
If  we  would  have  the  ratio  of  the  tension  q,  to  the  tension  h,  we 
should   multiply  only  the  two  first  proportions.     The  other  ratios 
may  be  found  in  a  similar  manner. 

If  it  were  proposed  to  determine  the  ratio  of  the  powers  among 
themselves,  we  have  only  to  deduce  from  the  above  series  of  ratios 
the  ratio  of  two  consecutive  powers  to  the  tension  of  the  same 
cord ;   thus, 

sin  FJE 
sin  IBG 
sin  IBH 
sin  KCL. 

Taking  the  product  of  the  corresponding  terms,  and   reducing,  we 

have 

p:r::s\ngAD  sin  GBH  sin  L  CM  :  sin  FAE  sin  IBH  sin  KCL. 

To  obtain  the  ratio  of  p  to  q,  we  should  multiply  only  the  terms  of 
the  two  first  proportions. 

It  will  hence  be  seen  how  we  ought  to  proceed  when  there  is  a 
greater  number  of  powers,  or  when  we  would  compare  the  tensions 
of  the  cords  with  the  powers  themselves. 

149.  If  the  powers  p,  q,  r,  bisect  the  angles  q  AB,  ABC, 
BC  TUT,  respectively,  the  angles  DAF,  FAE,  would  be  equal ;  and 
the  angles  GBH,  LCM,  would  have  the  same  sines  as  the  angles 
IBH,  MCK,  respectively  ;  whence,  by  means  of  the  above  ratios, 
it  will  be  seen  that  the  different  parts  of  the  cord  q  ABC  vj  would 
be  equally  stretched. 

150.  If  instead  of  the  powers^,  q,  r,  we  substitute  in  A,  B,  C, 
fixed  points  or  pivots,  the  pressure  upon  these  points  arising  from 
the  tension  of  the  extreme  parts  of  the   cords,   would  be   directed 

147.     in  such  a  manner  as  to  bisect  each  of  the  angles  ;  and   the   tension 

of  the  several  parts  of  the  cords  q  A,  AB,  &c.,  would  be  equal. 

Accordingly,  if  two  powers  q,  ts,  are  exerted  upon  a  cord  passing 
7\g.  59. 1'ound  the  periphery  of  a  polygon  or  of  any  curve,  the  tension  will 

communicate  itself  equally  to   every  part,  so  that  the   two  powers 

must  be  equal. 


Rope  Machine.  85 

151.  When  the  number  of  cords  united  by  the  same  knot 
exceeds  three,  being  in  the  same  plane,  or  when,  being  in  differ- 
ent planes,  the  number  exceeds  four,  the  directions  being  given, 
the  ratios  of  the  powers  and  of  the  tensions  of  the  cords  are  not 
absolutely  determinable  ;  that  is,  if  a  certain  number  of  powers 
(not  less  than  those  just  stated)  be  in  equilibrium  according  to 
known  directions,  we  can  substitute  instead  of  them  a  like  number 
of  other  powers  directed  in  the  same  manner,  but  which,  having 
very  different  ratios  among  themselves,  are  notwithstanding  in 
equilibrium.  If,  for  example,  the  four  cords  ^  p,  A  q,  Jl  r,  A  t;^,^^^-  60. 
are  all  in  the  same  plane,  and  having  taken  AB  to  represent  the 
force  p,  and  having  produced  the  cord  ^  A  to  C,  we  suppose  the 
effort  AB  composed  of  two  others  AC,  AD,  the  first  of  which  is 
equal  and  directly  opposite  to  the  power  cr,  nothing  can  be  inferred 
from  the  direction  AD  oi  the  action  that  is  to  oppose  itself  to  the 
effort  of  the  two  powers  q,  r ;  nothing,  I  say,  can  be  inferred 
from  this  direction,  except  that,  produced,  it  must  pass  into  the 
angle  q  Ar  ',  a.  condition  which  may  evidently  be  satisfied  in  an 
infinite  number  of  ways.  Accordingly,  if  AD  be  drawn  in  any 
manner  whatever,  within  the  angle  formed  by  A  q  and  A  r  pro- 
duced, and  we  construct  upon  AB  as  a  diagonal,  and  upon  the 
directions  AC,  AD,  as  sides,  the  parallelogram  ACBD,  and  then 
upon  ./5D  as  a  diagonal,  and  upon  q  A,  r  A,  produced,  as  sides, 
w^e  construct  also  the  parallelogram  AEDF,  AB  being  taken  to 
represent  the  value  of  p,  AC  may  be  taken  to  represent  that  of 
To,  AF  that  of  r,  and  AE  that  of  q.  This  is  evident,  because 
the  force  AB  is  equivalent  to  the  two  forces  AC,  AD,  the  first  of 
which,  in  order  to  be  in  equilibrium  with  cr,  must  be  equal  to  ar, 
and  the  second  AD  is  equivalent  to  die  two  forces  AF,  AE,  which, 
to  be  in  equilibrium  with  r  and  q,  must  be  equal  to  r  and  q  respec- 
tively. But  it  will  be  seen  at  the  same  time,  that  by  giving  to 
AD  a  different  direction,  AC,  AF,  AE,  will  have  different  values, 
but  such  notwithstanding  diat,  being  taken  to  represent  the  powers 
acting  in  these  directions,  an  equilibrium  would  be  produced  ;  so 
diat  in  this  case,  the  directions  remaining  the  same,  there  is  an  infi- 
nite variety  of  ways  in  which  an  equilibrium  can  be  effected  among 
the  powers  in  question. 

J  52.  The  problem   is  of  a  similar  character  when  the  cords 
proceeding  from  the  same  knot,  arc  in  different  planes,  and  amount 


86  Statics. 

to  more  than  four.  But  if  the  number  does  not  exceed  four,  the 
directions  being  given,  the  ratios  that  must  exist  among  the  forces 
applied  to  these  cords  respectively,  are  determinate.  For  through 
Fig-  61.  any  two  of  these  cords,  ns  A  p,  A  zr,  a  plane  may  be  supposed  to 
pass,  which  produced  would  meet  the  plane  r  Aq  o(  the  two 
other  cords  in  some  line  DAE,  the  position  of  which  is  determin- 
ed by  the  directions  of  the  four  powers.  Then,  the  direction  cr  A 
being  produced,  and  AB  being  taken  to  represent  the  power  p, 
if  upon  ^B  as  a  diagonal,  and  upon  the  directions  AD,  AC,  as 
sides,  we  construct  the  parallelogram  DACB,  AC  will  represent 
the  value  of  the  power  zu,  and  AD  the  effort  made  by  the  power 
p  against  the  two  powers  q  and  r  acting  conjointly.  Accordingly, 
having  produced  q  A  and  r  A  (which  are  in  the  same  plane  with 
AD)  to  F  and  G,  if  upon  AD  as  a  diagonal,  and  upon  AF,  AG, 
as  sides,  we  construct  the  parallelogram  AFDG,  AF,  AG,  will 
represent  the  values  belonging  to  the  two  powers  q  and  r. 

153.  Finally,  whatever  the  case  may  be,  whether  the  cords 
are  in  the  same  plane  or  not,  as  a  state  of  equilibrium  requires 
that  each  knot  should  remain  immovable,  if  the  force  or  tension 
of  each  cord,  applied  to  the  same  knot,  be  decomposed  into  three 
other  forces  parallel  to  three  rectangular  co-ordinates,  it  is  ne- 
cessary with  respect  to  each  knot  that  the  sum  of  the  forces  paral- 
lel to  each  of  these  lines  should  be  equal  to  zero  ;  (it  being  well  un- 
129.  derstood  that  by  the  word  sum,  as  here  used,  is  meant  the  sum  of 
the  forces  that  act  in  one  direction,  minus  the  sum  of  those  which 
act  in  the  opposite  direction.)  If  the  cords  united  by  the  same 
knot  were  in  the  same  plane,  it  would  be  sufficient  to  decompose 
the  tensions  respectively  into  two  forces  parallel  to  two  lines 
perpendicular  to  each  other,  and  drawn  in  the  same  plane.  This 
method  would  give  in  every  case  all  the  conditions  of  equilibrium, 
the  cords  being  supposed  to  be  firmly  connected  among  them- 
selves. 

To  give  a  simple  example  of  this  method,  let  it  be  proposed  to 
find  the  ratio  of  three  powers  in  equilibrium  by  means  of  three 
cords  united  by  the  same  knot. 

_.     .,        Let  us  suppose  for   a  moment  that  these  three  powers  admit 
"of  being  represented  by  the  three  lines -46*,  AB,  AF,  and  in   or- 
der to  abridge  the   decomposition,   let   the  two   powers  p,  q,  be 


Rope  Machine.  87 

decomposed  in  the  manner  indicated  by  the  figure,  that  is,  each 
into  two  parts,  one  in  the  direction  of  p,  and  the  other  perpendicu- 
lar to  this  direction.  Then  in  the  right-angled  triangles  BAC, 
FAJ,  we  shall  have,  radius  being  unity, 

BC=zAD  =  AB  sm  g  AC,  Trig.so. 

FI=AE=AF  smrAC, 

AC=AB  cos  q  AC, 

AI  =AF  cos  r  AC. 

Therefore,  according  to  the  principle  above  referred  to,  we  shall 
obtain,  ^-^• 

AB  sm  q  AC  — AF  sm  r  AC  =  0, 

and  ABcosq  AC -{-AF  cos  r  AC— AG  =0. 

The  first  of  these  equations  gives 

AB  sin  qAC  =  AF  sin  r  AC, 
and  consequently, 

AB  :  AF  ::  sm  r  AC  :  sm  q  AC, 
that  is, 

q  :  r  :  :  sm  r  AC  :  sm  q  AC, 

which  agrees   with  what  was  before  demonstrated.     If  the  value     143. 
of  AF  be  deduced  from  the  first  of  the  above  equations,   and  sub- 
stituted in  the  second,  we  shall  have 

Ajy  Ar^   \    AB  s\n  q  AC  COS  r  AC  .^        _ 

AB  cos  q  AC  -{ '-. j-pi AG  =  0, 

•'  sm  r  AC 

or 

.45  sin  r  ^Ccos  7 -4C  +  .4^  sin  g' ^C cos  r  ^C  =  ^G  sin  r^C. 

But 

sin  r^C  cos  q  AC  -\-  sin  q  AC  cos  r  AC  =■  sin  [r  AC  -\-  q  -4C),  Trjg.n. 

z=z  sm  q  A  r  ) 

therefore, 

AB  sm  q  A  r  =z  AG  sin  r  AC; 

that  is, 

AB   :   AG   :  :  sin  r  AC  :  sin  q  A  r. 


88  Statics. 

or, 

q   '.  p    :  :   ^\n  r  AC  :   s\n  q  A  r, 

143.     which  agrees  also  with  the  proposition  above  referred  to. 

154.  We  shall  now  inquire  into  the  changes  that  take  place  in 
the  communicalion  of  the  action  exerted  by  the  powers  in  con- 
sequence of  the  gravity  of  the  cords. 

Let  there  be  any  number  of  powers  applied  to  the  same  cord 
Fig.  63.  Q  ABC  xs  drawn  at  its  two  extremities  by  the  two  powers  §,  C7,  and 
retained  at  two  fixed  points  q  and  sr. 

If  we  produce  the  two  extreme  cords  q  A,  to  C^  until  they 
meet  in  T^,  it  is  evident  that  the  resultant  of  the  particular  tensions 
43.  of  these  two  cords  must  pass  through  the  point  J^.  And  since  an 
equilibrium  is  supposed,  the  resultant  of  the  three  powers  p,  q,  r, 
and  of  the  tensions  of  the  two  intermediate  cords  AB,  BC,  must 
also  pass  through  the  point  T^;  since,  in  order  to  an  equilibrium,  this 
resultant  must  be  equal  and  directly  opposite  to  the  resultant  of  the 
tensions  of  the  two  cords  q  A,  -as  C.  But  the  resultant  of  the 
three  powers,  and  of  the  tensions  of  the  two  intermediate  cords, 
is  nothing  but  the  resultant  of  the  three  powers  simply,  because 
each  of  the  two  cords  AB,  BC,  has  by  itself  no  action  whatevei", 
and  consequently  no  effect  upon  any  part  of  the  system.  There- 
fore the  resultant  of  all  the  powers  p,  q,  r,  applied  to  the  cord, 
passes  through  the  point  of  meeting  T^  of  the  two  extreme 
cords. 

42,  die.  It  has  been  shown  how  this  resultant  may  be  determined  ;  but 
if  the  cords  are  parallel,  as  is  the  case  when  the  powers  p,  q,  r,  are 
weights,  since  their  resultant  cannot  but  be  parallel  to  them,  its  di- 
rection is  found  very  simply  by  drawing  through  the  point  V^  a  line 
parallel  to  one  of  the  directions  of  these  weights,  that  is  by  drawing 
a  vertical  or  perpendicular  line. 

Accordingly,  let  there  be  any  number  of  weights  applied  to  the 
Fig.  64.  same  cord  q  ABCD  a  destitute  of  gravity.  The  two  extreme 
cords  being  produced,  and  a  vertical  VX.  being  drawn  through  their 
point  of  meeting,  we  can  reduce  the  equilibrium  of  the  whole  sys- 
tem to  the  case  in  which  the  three  powers,  applied  to  three  cords, 
are  united  by  the  knot  V,  and  in  which  the  power  directed  accord- 
ing to  XVZ  is  the  sum  of  the  weights.     We  conclude,  therefore, 


Rope  Machine.  89 

that  the  tension  q  is  to  the  tension  ct,  as  the  sine  XV  m  is  to  the 
sine  of  q  VX^  143, 

If  a  heavy  cord  now  be  considered  as  an  infinite  number  of 
small  weights  uniformly  distributed  along  the  axis  of  this  cord,  it 
will  be  seen,  that  if  rs  represent  the  point  where  the  power  is  applied  Fig.  65. 
to  the  cord,  and  q  that  in  which  this  cord  is  attached  to  a  machine, 
the  action  exerted  by  the  power  upon  the  point  q  will  be  transmitted 
in  the  direction  q  V  oi  ?i  tangent  to  the  curve  representing  the  fig- 
ure which  the  cord  assumes  by  the  action  of  gravity.  This  action 
is  not  equal  to  that  of  the  power  -uj,  except  when  the  vertical,  drawn 
through  the  point  of  meeting  V  of  the  two  extreme  tangents,  bisects 
the  angle  q  V  ^',  and  in  general  the  action  of  the  power  -a,  namely, 
that  which  it  would  transmit,  if  the  cord  were  destitute  of  weight, 
is  to  that  which  it  transmits  in  conjunction  with  the  weight  of  the 
cord,  as  the  sine  of  q  VX  is  to  the  sine  of  n;  VX. 

155.  We  remark,  that  strictly  speaking,  whatever  force  is  em- 
ployed to  stretch  a  cord  q  w,  this  cord  can  never  be  made  perfectly 
straight,  except  it  be  in  a  vertical  position  q  -us .  Let  us  suppose  the 
cord  r  Ap,  destitute  of  gravity,  to  support  the  weight  q,  by  means  Fig.  66. 
of  the  two  equal  powers  p,  r,  the  directions  of  which  are  such 
as  to  form  an  angle  approaching  infinitely  near  to  180°,  we  shall 
have 

q  :  p  ::  sin  CAD  :  sin  CAB  ; 

or,  DA  being  produced, 

q  :  p  : :  sin  CAS  :  sin  i  CAD  ; 

but  the  angle  CAS  is  by  supposition  infinitely  small,  and  i  CAD 
approaches  infinitely  near  to  a  right  angle ;  therefore  q  must  be 
infinitely  small  with  respect  top  ;  and  even  where  the  weight  5-  is 
infinitely  small,  the  two  parts  of  the  cord  still  make  an  angle  with 
each  other,  and  are  not,  strictly  speaking,  in  the  same  straight  line. 

It  may  hence  be  inferred,  that  a  very  small  force  q  will  cause  a 
very  great  tension  in  the  cords  A  p,  A  r,  when  the  angle  r  A  p 
formed  by  them  is  very  obtuse. 

We  are  able,  also,  upon  the   same  principle,  to  explain   why,  in 
blowing  through  a  tube  A  a,  into  a  flexible  bag  a  EB  C  a,  the  ex-  Fig.  67. 
Mech.  12 


143. 
Trig.  13. 


154 


90  Statics. 

tremity  B  ol  which  is  ntlachcd  lo  a  weight  p,  wc  are  able,  1  say,  to 
explain,  why  a  moderate  impulse  of  the  breath  suffices  to  raise  the 
weight  p^  although  very  considerable.  Indeed,  each  half  a  EB, 
a  CB,  of  the  vertical  section  of  this  bag  may  be  considered  as  a 
cord,  pressed  at  each  point  by  a  perpendicular  force  equal  to  that 
exerted  by  the  air.  The  resultant  of  all  these  pressures  must  be 
directed  according  to  FED,  that  is,  it  must  pass  through  the  point 
of  meeting  of  the  tangents  belonging  to  the  extremities  of  this 
cord,  and  must  be  to  the  effort  made  in  the  direction  BD,  as  the 
sine  of  a  DB  or  of  a  D  u,  is  to  the  sine  of  FD  a.  Now  the 
angle  a  D  u  \s  very  small.  Therei'bre  a  very  small  effort  in  the 
direction  jPD  produces  a  very  great  effect  in  the  direction  BD; 
and  accordingly  the  pressure  exerted  upon  a  EB  will  cause  a  con- 
siderable effort  in  the  direction  BF,  and  the  weight  will  be  drawn 
by  two  forces  of  considerable  magnitude  in  the  direction  BD,  BF, 
which  will  have  so  much  the  greater  effect,  according  as  the  angle 
FBD  is  smaller,  since  their  resultant  will  approach  so  much  the 
50.    nearer  to  the  sum  of  the  components. 


Of  the  Lever. 

156.  By  the  lever,  we  understand  an   inflexible  rod,  of  any  fig- 
^'S-  ^p  lire  whatever,  so  fixed  at  some  point  F,  as  to  admit  of  no  other  mo- 
tion, by  the  action  of  the  forces  that  are  applied  to  it,  but  a  motion 
of  rotation,  that  is,  a  motion,  by  which  it  turns  about  the  fixed  point 
F.     This  point  is  called  the  fulcrum. 

We  first  consider  the  lever  as  an  inflexible  line  without  mass  and 
without  gravity.  In  the  case  of  an  equilibrium,  we  can  easily  make 
allowance  for  the  gravity  of  the  parts,  by  supposing  it  collected  at 
the  centre  of  gravity  of  this  lever,  and  thus  regarding  it  as  a  new 
force  applied  at  this  point  according  to  a  vertical  direction.  In 
case  of  motion,  it  is  not  at  the  centre  of  gravity  that  we  are  to  sup- 
pose the  mass  collected,  but  at  some  other  point  to  be  determined 
hereafter. 

We  shall  proceed  on  the  supposition,  that  the  forces  applied  to 

the  lever,  are  all  in  the   same  plane   with   the  fulcrum.  We  shall 

treat  in  another  place  of  equilibrium  and   motion   when  the  forces 
applied  to  the  lever  are  in  different  planes. 


Lever,  9 1 

157.  Let  there  be  two  forces  or  powers^,  9,  applied  at  the  two 
points  B,  D,  of  the  lever,  BFD,  either  immediately,  or  by  means 
of  two  cords,  or  two  rods  without  mass,  acting  upon  this  lever 
in  the  directions  B  p,  D  q,  and  being  in  equilibrium.  It  is  pro- 
posed to  determine  the  conditions  of  this  equilibrium. 

As  one  of  the  two  powers,  q  for  example,  cannot  be  in  equili- 
brium with  the  other  except  by  means  of  the  fulcrum  F,  it  is  evi- 
dent that  the  power  q  must  produce  two  efforts,  one  of  which  anni- 
hilates that  of  the  power  p,  and  the  other  is  destroyed  by  the 
fulcrum  F,  and  consequently  passes  through  this  point. 

Let  the  lines  p  B,  q  D,  representing  the  directions  of  the  pow- 
ers, be  produced  till  they  meet  in  some  point  A,  and  join  AF. 
The  power  q  may  be  supposed  to  be  applied  at  A,  according  to 
A  q;  then  if  AG  represent  the  value  or  magnitude  of  this  power, 
and  upon  AG,  as  a  diagonal,  and  in  the  directions  AF,  BAE,  as 
contiguous  sides,  we  construct  the  parallelogram  AHGE;  AE 
will  represent  the  effort  made  by  the  power  q,  according  to  this 
line,  and  in  a  direction  opposite  to  that  of  p ;  and  AH  will  be  that 
exerted  against  the  fulcrum  F.  Indeed,  although  the  point  A  is 
not  connected  with  the  two  points  B,  F,  the  force  q  is  distributed 
in  the  same  manner  as  if  A  were  thus  connected.  For  it  is  evident, 
that  if,  without  changing  the  forces  or  their  directions,  we  connected 
the  point  A  with  the  three  points  B,  F,  D,  by  means  of  three 
inflexible  rods  AB,  AF,  AD,  without  mass,  this  would  not  alter  in 
any  degree  the  supposed  state  of  the  system  or  the  manner  in  which 
the  force  q  is  exerted.  Now  in  this  last  case,  the  action  of  the  force 
q  would  manifestly  be  communicated  in  the  manner  we  have  men- 
tioned ;  therefore  it  would  be  communicated  in  the  same  manner, 
according  to  the  first  supposition.  This  being  established,  in  order 
that  there  may  be  an  equilibrium,  it  is  necessary  that  the  force  AE 
should  not  only  have  a  direction  contrary  to  that  of  the  force  jh  but 
that  it  should  also  be  equal  to  j).  As  to  the  force  AH,  in  order 
that  it  may  be  destroyed,  it  is  sufficient  that  it  be  directed  to  the 
point  F.  Accordingly,  if  we  designate  the  force  exerted  against  the 
fulcrum  by  ^,  we  shall  have 

q   :  p  :  Q  ::  AG   :  AE  :  AH 


43. 


92  Statks. 

158.  If  from  A  towards  B  we  take  AI=  AE,  and  join  IH, 
AIHG  will  be  a  parallelogram.  But  Al,  AG,  the  sides  of  this 
parallelogram,  represent  the  magnitudes  and  directions  of  the  two 
forces  p,  q,  consequently  the   diagonal  AH  represents  their  result- 

38.  ant  ;  therefore,  since  AH  thus  represents  the  force  exerted  against 
the  fulcrum,  it  may  be  inferred  as  a  general  conclusion  that  the 
force  exerted  against  the  fulcrum  is  precisely  the  resultant  of  the 
two  forces  applied  to  the  lever ;  and  that  consequently  these  two 
forces  act  against  the  fulcrum  as  if  they  were  immediately  applied 
to  it  according  lo  directions  respectively  parallel  to  those  in  which 
14^-     they  are  actually  exerted. 

Indeed,  this  last  truth  may  be  rendered  evident,  by  observing 
that,  instead  of  the  force  q,  may  be  substituted  the  two  forces  AE, 
AH,  the  first  of  which  is  destroyed  by  the  force  p,  and  the  remain- 
ing force  AH  is  the  single  efTect  to  which  the  two  forces  p  and  q 
are  reduced,  and  by  consequence  the  resultant  of  these  two 
forces. 

159.  By  means  of  the  ratios 

q:  p  :  Q  ::  AG  :  AE:  AH, 

15-  above  found,  we  are  able  to  compare  the  forces  q  and  p,  as  well 
with  each  other,  as  with  the  force  exerted  against  the  fulcrum. 
But  as  this  ratio  is  not  the  most  convenient,  we  proceed  to  find  two 
others  which  may  be  employed  for  the  same  purpose. 

48.  (1.)  According  to  a  principle  already  established,  we  have 

AG  '.AE:  AH : :  sin  HAE  :  sin  HAG  :  sin  GAE, 

or, 

: :  sin  HAI  :  sin  HAG  :  sin  GAI, 

since  the  angles  HAE,  GAE,  have  the  same  sines  respectively  as 
Trig.  13.  their  supplements  HAI,  GAI ;  that  is,  the  forces,  q,  p,  q,  are  each 
represented  by  the   sine   of  the   angle  comprehended  between  the 
directions  of  the  two  others. 

(2.)  It  has  been  shown  that  with  respect  to  three  forces  of 
which  one  is  the  resultant  of  the  two  others,  either  two  are  always 
to  each  other  reciprocally  as  tlie  perpendiculars  let  fall  upon  their 
directions  from  any  point  taken  in  the  direction  of  the  third.     Ac- 

49.  cordingly,  if  from  any  point  in  AF,  as  F,  for  example,  we  let   fall 


Lever.  93 

the  perpendiculars  FZ,,  FM,  upon  the  directions  p  B,  q  D,  we 
shall  have 

q   :  p   ::  FL   :  FM. 

In  like  manner,  if  from  any  point  in  the  direction  of  the  force 
gr,  as  D  for  example,  we  let  fall  the  perpendiculars  DO,  D  g,  upon 
the  directions  of  the  force  p,  and  q,  we  shall  have 

p   :  g   ::  Bg   :   DO. 

The  force  q  may  also  be  compared  in  the  same  way  with  the 
force  g  exerted  against  the  fulcrum. 

All  these  propositions  hold  true,  whatever  be  the  form  of  the 
lever,  and  whatever  be  the  directions  of  the  two  powers   employed. 

160.  When  the  directions  of  the  two  powers  are  parallel,  in 
which  case  the  resultant,  or  force  exerted  against  the  fulcrum,  is 
parallel  to  them,  the  perpendiculars,  let  fall  from  the  same  point  ^^• 
in  the  direction  of  one  of  these  forces,  upon  the  directions  of  the 
other  two  respectively,  are  all  in  the  same  straight  line  LFM.  ^'S*  '^^' 
We  may  say,  therefore,  in  this  case,  having  drawn  the  line  LFM 
perpendicular  to  the  direction  of  the  powers,  that  each  force  or 
power  is  represented  by  the  part  of  this  straight  line  comprehended 
between  the  directions  of  the  two  others. 

1 G 1 .  If,  moreover,  the  lever  is  straight,  it  will  follow  from  the 
circumstance  of  the  triangles  FLB,  FMD,  being  similar,  that  the 
parts  FB,  FD,  BD,  have  the  same  ratio  to  each  other  as  the  parts 
FL,  FM,  LM;  we  may  say,  therefore,  in  this  case  that  each  force  Geom. 
IS  represented  by  the  part  of  the  lever  comprehended  between  the 
directions  of  the  two  others. 

Thus, 

q    :  p   ::   FB    :   FD; 

that  is,  the  powers  are  to  each  other  in  the  inverse  ratio  of  the  two 
arms  of  the  lever  FB,  FD ;  so  that  die  power  q,  in  order  to  be  in 
equilibrium,  must  be  as  much  smaller  than  p,  as  the  arm  to  which  it 
is  applied  is  longer  than  the  arm  to  which  p  is  applied.  As  to  the 
force  g  exerted  against  the  fulcrum,  it  is  equal  to  the  sum  of  the 
two  powers  q,  p  ;  since  these  being  represented  by  FD,  FB,  the 
former,  or  g,  is  represented  by  BD. 


94  Statics. 

Fig.  71,  162.  If  we  make  a  distinction  in  the  forces  or  powers,  by 
'  ■  regarding  one,  q  for  instance,  as  giving  motion,  j»  as  receiving  it,  and 
JP  as  a  pivot  or  point  of  support,  we  may,  in  the  manner  of  the 
ancients,  make  three  sorts  of  levers,  according  to  the  three  different 
situations  in  which  the  agent  q  can  be  placed  with  regard  to^  and 
jP.  Figure  71  represents  what  is  called  a  lever  of  the  first  kind, 
in  which  the  agent  and  the  resistance  are  on  opposite  sides  ol  the 
fulcrum,  and  the  agent  will  have  so  much  the  more  advantage 
according  as  its  distance  from  the  fulcrum  is  greater  than  that  of 
the  resistance.  Figure  72  represents  a  lever  of  the  second  kind,  in 
which  the  resistance  is  between  the  agent  and  the  fulcrum,  and 
which  consequently  is  always  favorable  to  the  agent.  Figure  73 
represents  a  lever  of  the  third  kind,  in  which  the  agent  is  between 
the  resistance  and  the  fulcrum  ;  in  this  case,  therefore,  the  power  of 
the  agent  is  always  employed  to  disadvantage  ;  and  such  a  lever  is 
never  to  be  used,  where  the  object  is  to  augment  the  effect  of  the 
agent;  that  is,  where  it  is  proposed  to  overcome  a  greater  force. 
But  as  the  purpose  to  be  fulfilled  is  not  always  to  increase  the  power 
of  the  agent,  this  circumstance  does  not  prevent  this  third  kind  of 
lever  being  very  usefully  employed  in  machinery,  where  we 
would  avail  ourselves  of  every  species  of  motion  that  we  can  dispose 
of.  Thus  in  turning,  in  weaving,  in  spinning,  and  in  various  kinds 
of  manufacture,  where  great  velocity,  and  not  great  force  is  required, 
and  where  the  hands  of  the  laborer  are  occupied  with  the  more 
important  parts  of  the  work,  this  species  of  lever  is  adopted  with 
obvious  advantage,  the  feet  being  employed  to  give  motion  to  the 
machinery. 

163.  Before  proceeding  farther,  we  will  observe,  that  setting 
aside  friction,  the  fulcrum  is  not  to  be  considered  as  simply  a  pivot, 
Fig.  69.  or  support.  Indeed,  if  the  fulcrum  F,  instead  of  penetrating  into 
the  interior  of  the  lever,  as  represented  in  the  figure,  only  touched 
the  surface,  it  is  evident  that,  although  the  two  powers  q,  p,  were  in 
the  inverse  ratio  of  the  distances  of  the  perpendiculars  FAl,  FL, 
they  would  still  not  be  in  equilibrium,  except  in  the  single  case, 
where  the  direction  AF  is  perpendicular  to  BD  (  or  to  the  tangent 
at  F  in  figure  68)  ;  for,  if  AF  were  oblique,  it  would  clearly  tend 
to  communicate  motion  to  the  lever  in  the  direction  BD.  Thus,  we 
should  err  in  supposing,  for  example,  that  (friction  and  the  gravity 


Lever.  95 

of  the  lever  being  left  out  of  the  question)  the  two  weights  p  and  q, 
would  remain  in  equilibrium  in  the  inclined  position  represented,  if,  Fig- 74. 
p  being  to  5'  as  JP*;^  to  Fp,  the  surface  of  the  lever  merely  rested 
upon  the   point  F.     The  fulcrum,  in  order  that  there  may  be  an 
equilibrium  in  all  positions  of  the  lever,  must  have  the  effect  of  a 
pin  passing  through  it.     In  short,  when  we  say,  it  is  sufficient,  that 
the  resultant  AF  of  the  two  powers  should  pass  through  the  fulcrum 
F,  it  is  taken  for  granted  that  the  corresponding  point  F  of  the  lever 
does  not  admit  of  any  motion  ;  for  otherwise  this  condition  is  not 
sufficient.    For  example,  if  the  lever  BD  were  drawn  by  three  forces 
p,   q,   r,  applied  to    the  three  chords  B  p,  D  q,  F  r,  there  would  F'&-  '^^' 
not   be  an  equilibrium,    if  AF  were  the  direction  of   the  resultant 
of  p  and  q,  although  it  should   pass  through  the  point  F ;  it  would 
be    further    necessary    that    the    point   of  meeting  A  should  be    &i. 
inr  F. 

164.  Since  the  two  forces  p,  q,  in  equilibrium  by  means  of  the 
lever  BFD,  must  be  in  the  inverse  ratio  of  the  perpendiculars  FL,  Fig.  68, 
FM;  that  is,  since  it  is  necessary  thatp  should  be  to  q,  as  FM  to     ' 
FL,  it  follows  that  p  X  FL  =  q  X  FM ;  in  other  words,  the 
moments  of  the  two  forces,  taken  with  respect  to  the  fulcrum,  or 
any  other  point  in  the  direction  AF,  must  be  equal.  ^• 

1G5.  As  there  cannot  be  a  force  without  a  tendency  to  motion, 
by  the  forces  p,  q,  is  to  be  understood  the  product  of  a  certain  mass 
by  the  velocities,  that  these  forces  would  respectively  communicate 
to  this  mass,  if  it  were  free.  Thus  let  m  be  a  certain  mass,  and 
u  the  velocity  that  the  force  p,  acting  freely,  is  capable  of  giving  it  j 
also,  let  n  be  another  mass,  and  v  the  velocity  that  the  force  q  is 
capable  of  giving  it ;  in  order  that  there  may  be  an  equilibrium,  the 
following  proportion  is  necessary  ;  namely, 

m  X   u  :  n   X   v  ::  FM  :  FL. 

IGG.  Let  w  be  the  velocity  produced  by  the  force  of  gravity  in 
an  instant ;  and  let  m,  n,  be  two  heavy  bodies  attached  to  two  cords  Fig.  76. 
BI  m,  DK  n,  which  passing  over  two  round  bodies  1,  K,  transmit 
entirely  to  the  lever  BFD,  according  to  any  proposed  directions 
Bl,  DK,  the  action  of  gravity  of  these  bodies  ;  we  shall  have  10  in, 
V3 11.  as  the  measures  of  the  forces  with  which  these  bodies  act  re- 


96  Statics. 

27.      spectively  ;  we  must  have,  therefore,  in  order  that  there  may  be  an 
equilibrium, 

wm  :  wn  :  :  FE  :  FC, 
that  is, 

m  :    n     :  :  FE  :  FC; 

therefore,  in  order  that  there  may  be  an  equilibrium  between  two 
bodies  which  are  urged  only  with  the  force  of  gravity,  or  between 
any  two  bodies  that  tend  to  move  with  equal  velocities,  it  is  suffi- 
cient that  the  masses  of  these  bodies  be  in  the  inverse  ratio  of  the 
distances  of  their  directions  from  the  fulcrum. 

1G7.  But  if  the  velocities  with  which  the  bodies  tend  to  move, 
be  unequal,  it  is  not  the  masses,  but  the  products  of  the  masses 
into  the  velocities,  which  must  be  in  the  inverse  ratio  of  the  distances 
of  their  directions  from  the  fulcrum. 

168.  If  two  finite  and  heavy  masses  m,  w,  are  urged  by  finite 
and  unequal  velocities,  according  to  the  directions  I m,  Kn;  as 
the  velocity  which  gravity  is  capable  of  giving  in  an  instant  (or 
infinitely  small  portion  of  time)  is  infinitely  small ;  in  order  that 
the  two  finite  velocities  may  mutually  destroy  each  other,  it  is 
sufficient  that  the  quantities  of  motion  which  the  two  bodies  would 
have  in  virtue  of  these  velocities  should  be  in  the  inverse  ratio 
of  FE,  FC.  But  this  equilibrium  would  not  exist  except  for 
an  instant;  for  when  these  velocities  are  mutually  destroyed,  the 
bodies  m,  n,  subjected  to  the  action  of  gravity,  would  receive  quan- 
tities of  motion,  which  would  be  in  the  simple  ratio  of  the  masses, 
and  which  consequently  would  no  longer  be  in  the  inverse  ratio 
of  the  distances  FE,  FC. 

We  hence  see  the  difference  between  an  equilibrium  among 
bodies  urged  by  gravity  only,  and  an  equilibrium  among  bodies 
urged  by  unequal  finite  velocities. 

It  may  be  remarked,  moreover,  that  it  is  impossible  to  put 
in  equilibrium  a  body  urged  by  gravity  only  with  a  body  urged 
by  a  finite  velocity ;  and  we  may  hence  conclude,  that  if  the 
Fig  71. weighty  is  in  equilibrium  with  a  force  q  exerted  by  an  animal, 
this  last  does  not  tend  to  move  the  point  D,  except  with  a  velocity 
infinitely  small.      If  on  the  contrary  the  force  q,  applied  at  D, 


Lever.  97 

acted  by  means  of  a  blow  or  finite  impression,  it  would  raise  the 
weight  p,  however  great  it  might  be,  at  least  during  a  certain  time, 
which,  when  p  is  very  large,  may  be  such  that  the  eye  cannot  dis- 
tinguish it ;  but  the  motion  would  not  be  the  less  real.  This  sub- 
ject will  be  placed  in  a  clearer  light  hereafter,  when  we  come  to 
treat  of  Collision. 

169.  By  means  of  the  ratio  which  we  have  established  between     157. 
the  two  powers  p,  q,  and  the  force  exerted  against  the  fulcrum  jP,  Fi^.  68, 
we   shall   be   able   to  solve  this   general   question.      Three  of  these 

six  things,  namely,  the  two  powers,  the  force  exerted  against  the 
fulcrum,  and  the  three  directions,  being  given,  to  find  the  three 
others.  When,  however,  the  directions  only  are  given,  we  can 
merely  find  the  ratio  of  the  forces  p,  q,  q.  The  solution  in  this  144. 
case  is  evident  from  what  has  been  said.  It  may  be  easily  obtained 
also  by  geometrical  construction  upon  which  we  will  only  observe, 
that  when  the  directions  are  parallel,  the  question  is  solved  by 
articles  53,  160;  and  that,  in  general,  if  it  is  proposed  to  deter- 
mine the  position  of  the  fulcrum,  when  the  powers  p  and  q,  and 
their  position  are  known,  the  question  reduces  itself  to  finding,  by 
article  38,  the  resultant  of  these  two  powers. 

170.  The  problem  is  different  when  more  than  two  powers  are 
applied  to  the  lever ;  in  this  case,  as  in  that  of  the  cords  in  article 
151,  we  can  vary  without  end  the  ratio  or  the  directions  of  some  of 
the  powers,  the  others  remaining  the  same,  and  yet  not  destroy 
the  equilibrium.  There  is,  however,  this  difference,  between  the 
lever  and  cords,  that  the  condition  of  equilibrium  in  the  former  is 
single,  whereas  in  the  latter  there  are  as  many  conditions  of  equi- 
librium as  there  are  knots.  It  will  suffice  to  point  out  the  condition  153. 
of  equilibrium  in  the  lever,  when  three  powers  are  employed,  to 
make  it  evident  that  the  proposition  will  hold  true,  for  any  greater 
number  of  powers. 

171.  Let  the  three  powers  p,  q,  r,  directed  according  to  B  p,F\g.  77. 
E  q,  D  r,  be   in  equilibrium   by  means  of  the  lever  BFD.     The 
power  q  may   be  considered    as  exerted  in   part   against  each  of 

the  powers  p  and  r,*   and   in  part  against  the  fulcrum  F.     Having 

*  The  power  q  cannot,   strictly  speaking,   be  considered  as  ex- 
erted against  r,  since  they  both  tend  to  turn  the  lever  in  the  same 
Mech.  13 


98  Statics. 

produced  the  directions  B  p,  E  q,  and  taken  from  the  point  of  meet- 
ing A,  the  line  AH  to  represent  the  power  </,  we  decompose  this 
power  into  two  others,  one  AG  equal  and  directly  opposite  to  the 
power  p,  and  the  oilier  AC  such  as  will  admit  of  heing  in  equilihrium 
with  the  power  r,  hy  means  of  the  fulcrum  F.  Accordingly,  if  the 
direction  D  r  meet  -^4C  at  the  point  /,  we  may  suppose  the  force  AC 
33.  applied  at  /,  according  to  the  direction  ACIL  ;  the  force  AC  or  IL 
must  therefore  he  capahle  of  heing  decomposed  into  two  others,  one 
/&"  equal  and  directly  opposite  to  the  power  r,  and  the  other  IM  di- 
rected against  tlie  fulcrum  F.  Thus  the  force  q  produces  the  three 
effects  AG,  IK,  IM,  of  which  the  two  first,  being  equal  and  directly 
opposite  to  the  forces  p  and  r,  are  destroyed,  and  the  last,  being 
directed  against  the  fixed  point  F,  cannot  but  be  destroyed  also. 
Now,  since  all  the  forces  which  act  upon  the  lever,  are  p,  q,  r,  or 
AG,  IK,  IM,  and  AG,  IK  are  destroyed,  we  conclude  that  IM  is 
the  resultant  of  the  three  powers^,  q,  r,  and  that  consequently  the 
only  condition  necessary  to  an  equilibrium  is,  that  this  resultant 
should  pass  through  the  fulcrum  F.  We  see,  therefore,  that  the 
powers  p,  q,  r,  act  upon  the  fulcrum  as  if  they  were  immediately 
applied  to  it  according  to  directions  parallel  to  those  which  they 
actually  have ;  and  this  conclusion  would  hold  true  for  any  number 
whatever  of  powers,  for  we  may  always  suppose  one  of  the  powers 
to  he  in  equilibrium  with  all  the  others  by  means  of  the  resistance 
of  the  fulcrum. 

172.  Since  F  must  be  in  one  of  the  points  of  the  resultant,  it 
must  have  the  properties  of  which  mention  has  already  been 
63.  made  ;  that  is,  when  several  powers,  exerted  in  the  same  plane,  are 
in  equilibrium  by  means  of  a  lever  of  whatever  figure,  if  from  the 
fulcrum  we  let  fall  perpendiculars  vpon  the  directions  of  these  forces, 
and  multiply  each  force  by  the  corresponding  perpendicular,  in  other 
ivords,  if  wetalcc  the  moments  of  the  forces  with  respect  to  the  fulcrum, 
the  sum  of  the  moments  of  the  forces  which  tend  to  turn  the  lever  in 

direction  about  the  fulcrum.  The  two  powers  p,  q,  being  rep- 
resented by  AH,  and  GA  or  AG'  respectively,  and  being  exerted 
in  these  directions,  are  equivalent  to  AC  ox  IL,  the  direction  of 
which  is  opposed  to  r,  since  it  tends  to  turn  the  lever  in  the  con- 
trary direction  ;  and  the  resultant  of  IL  and  IK',  that  is,  of 
p,  q,  /•,  is  IM. 


Lever.  99 

one  direction,  must  be  equal  to  the  sum  of  the  moments  of  those  which 
tend  to  turn  it  in  the  opposite  direction  ;  which  may  be  expressed 
generally,  by  taking  with  contrary  signs  the  moments  of  the  forces 
which  tend  to  turn  the  lever  in  opposite  directions,  and  saying,  that 
the  sum  of  the  moments  must  be  zero. 

173.  Accordingly,   all  that   we    have  said   with   respect  to  the 
value  and  direction  of  the  resultant,  is  applicable  here  to   the  deter-     53. 
mination  of  the  force  exerted  against  the   fulcrum,  and  the   position 

of  this  point,  whatever  be  the  number  of  powers. 

174.  Knowing,  for  example,  the  two    weights/^  and  q,  together  Fig.  78. 
with  the  length  and  weight  BD  of  the  lever,  if  we  would  determine 

the  fulcrum  F,  upon  which  the  whole  would  remain  in  equilibrium, 
we  should  consider  the  weight  of  the  lever  as  a  new  force  r,  applied 
at  the  centre  of  gravity  G  of  the  lever,  and  it  would  be  necessary 
that  the  moment  of  p  with  respect  to  the  unknown  point  F  should 
be  equal  to  the  sum  of  the  moments  of  r  and  9,  taken  with  respect 
to  the  same  unknown  point  jP. 

Let  the  lever  BD  be  straight,  and  of  a  uniform  magnitude  and 
specific  gravity  ;  and,  bearing  in  mind,  that  on  account  of  the  direc- 
tions of  the  forces  being  parallel,  instead  of  the  perpendiculars  FL, 
FK,  FM,  we  may  employ  the  parts  BF,  FG,  FD,  which  have  the 
same  ratio  to  each  other,  we  shall  have 

p   X   BF  =  r  X   FG  +  q  X   FD. 

Let  a  be  the  length  of  the  lever,  x  the  distance  BF;  we  shall 
have, 

BG  =  ^a,         FG  =  ^a~x,         FD  =  a~x. 

Let  s  be  the  specific  gravity  of  the  lever,  or,  in  other  words,  the 
weight  of  each  inch  in  length  of  this  lever ;  a  and  x  being  also 
counted  in  inches  ;  s  a  will  be  the  whole  weight  r.  We  have 
accordingly, 

p  X  =  s  a  (^  a  —  x)  -{-  q  (a  —  x), 
=  is  a*  —  sax  -}-  q  a  —  q  x, 

from  which  we  obtain 

X  =^ 


j)-\-  sa-\-  q 


100  Statics. 

Let  a  =  24  inches,  p  =  20  pounds,  q  =  4  pounds,  s  =  -j'a  of 
a  pound  ;  we  shall  have, 

1    (24)2  I   4.24        24  +  96 
^  —      OQ  ^  2  _j_  4      —         20         ~-  ^  T3  i"t.ueb  , 

that  is,  the  fulcrum  F,  in  order  that  there  may  be  an  equilibrium, 
must  be  placed  4 -^^-j  inches  from  the  extremity  i?;  whereas,  by 
neglecting  the  weight  of  the  lever,  we  should  have 

a  a  96        ,  .     , 

""  —  =:--:=  4  mches. 


i>  +  2  -  24 

If,  on  the  other  hand,  the  point  B  and  the  point  F  were  given, 
and  it  were  proposed  to  find  the  point  D  where  the  force  q,  sup- 
posed to  be  known  as  well  as  p,  must  be  applied  to  produce  an 
equilibrium  ;  designating  BF  by  b,  and  BD  by  y,  the  equation  of 
the  moments  becomes 

pb  =  sy  {^ij  —  b)  -]r  q  {y  —  h) 
=  isy~  —  syb  +  qy  —  qh; 


whence, 


and 


(2g  —  2s6)i/_2p6-f2(/6 


. ,       .  sh  —  q^        Uq  —  sbf    ,    2  p  b  4-2  a  b 


Fig.  78. 


__  sb  —  q±:\/{(/  —  sbY-^(2pb+2qb)s 

s 

The  positive  value  of  y  in  this  result  gives  the  distance  BD  in 
figure  78,  and  the  negative  value  gives  the  distance  BD  in  figure 
79,  the  distance  BF  being  supposed  without  gravity. 

If  we  would  have  the  distance  or  length  y  at  which  the 
weight  of  the  part  FD  would  of  itself  be  sufficient  to  counterbal- 
ance the  weight  p,  we  should  put  q  =.  0,  which  reduces  the  above 
result  to 


864-^/82  62  +  2506 
y    =• ! ? £ 

''  S 


Lever.  101 

If,  knowing  p,  q,  BF,  and  the  specific  gravity  of  the  lever  DF,  Fig.  80. 
we  would  determine  the  distance  FD  at  which  the  power  q  must  be 
placed ;  designating  FD  by  y,  BF  by  b,  we  shall  have  s  y  for  the 
weight  r;   and,  accordingly, 

P^  +  isy^  =  qy, 

from  which  y  is  easily  obtained. 

In  figure  73,  it  is  evident  that  the  longer  the  lever  is,  the 
more  the  power  q  is  to  be  diminished,  till  it  becomes  zero,  after 
which  it  must  act  in  a  contrary  direction  to  produce  an  equili- 
brium. 

In  figure  80,  as  the  lever  is  increased  in  length,  the  power  q 
becomes  at  first  less  and  less  to  a  certain  point  beyond  which  it  begins 
to  augment.  This  may  be  easily  shown  in  several  ways,  and  among 
others  by  the  equation 

pb  -i-  isy^  =  qy, 
which  gives 

_p_M:JLil! 

by  which  it  will  be  seen,  that  when  y  =0,  q  must  be  infinite  ; 
and  that  when  y  is  infinite,  q  must  also  be  infinite.  Accordingly, 
between  these  extremes  the  values  of  q,  must  be  finite,  and  there 
must  be  some  point  where  it  will  be  the  smallest  possible.  In 
order  to  determine  this  point,  we  have  merely  to  put  equal  to  q^j  ^^ 
zero  the  differential  of  the  value  of  q,  taken  by  regarding  y  only 
as  variable  ;  we  have  thus  C^^i  ^ 

_iP±±A±y^Ul^sdy  =  0', 

whence 

sy^'  =  pb  -{-  ^s  y^, 
and 


Therefore  the  value  of  the  smallest  power  q,  which  can  be  employed 
with  a  heavy  lever,  of  the  second  kind,  is 


102  Statics, 

— ^     or      V^i'  s  & 


J^ 


and  the  length  of  this  lever  is 


J 


2  p  b 


It  will  hence  be  perceived,  that  when  a  weight  is  to  be  raised  by 
a  heavy  lever,  employed  as  in  figure  81,  a  particular  length  is  neces- 
sary in  the  lever,  in  order  that  the  force  may  act  to  the  greatest 
advantage,  and  that  a  given  effect  may  be  produced  with  the  least 
possible  force ;  and  that  a  greater  or  less  length  would  be  attended 
with  a  loss  of  power.  There  is  accordingly  a  difference  in  this 
respect  between  a  heavy  lever  and  a  lever  without  weight. 


Of  the  Pulley. 

175.  A  pulley  is  a  solid  circle  or  wheel  having  a  groove  formed 
round  its  circumference,  and  an  axis  passing  perpendicularly  through 
its  centre,  and  through  a  case  or  frame-work  called  the  block. 
The  several  parts  taken  together,  are  sometimes  called  the  block, 
and  sometimes  simply  the  pulley. 

The  different  kinds  of  pulleys  may  be  reduced  to  two,  the 
fixed,  and  the  movable. 

The  fixed  pulley  is  that  in  which  the  power  and  the  weight  (or 
83.         resistance  to  be  overcome)   are  both  applied  according  to  directions 
that  are  tangents  to  the  circumference  of  the  pulley. 

-p.    ^         In  the  movable  pulley,  the  weight  or  resistance   is  applied  at 
85, 86.    the  centre,  or  in  a  direction  passing  through  the  centre  or   axis  of 
the  pulley. 

This  machine,  considered  in  a  general  point  of  view,  is  suscep- 
tible of  two  sorts  of  motion  ;  one  by  which  the  rope  passing  through 
the  groove  of  the  pulley,  changes  its  place  without  altering  the  posi- 
tion of  the  body  of  the  pulley,  the  other  is  such  that  the  body  of 
the  pulley  changes  its  situation  at  the  same  time.  Thus  a  state  of 
equilibrium  requires  two  different  conditions.  The  first  is  that  the 
two  parts  of  the  rope  which  embraces  the  pulley  should  be  equally 


Pulley.  103 

stretched,  and  thus  mutually  destroy  each  other.  The  second  con- 
dition is  derived  from  the  first  in  the  following  manner  ; 

176.  From  the  tension  of  the  two  parts  of  the  rope  which  passes 
over  the  pulley,  there  results  an  effort  upon  the  body  of  the  ma- 
chine which  may  be  determined  by  taking  in  the  directions  of  the 
ropes,  beginning  at  their  point  of  meeting,  LS.,  IB,  equal  to  each 
other,  and  forming  the  parallelogram  LdDB,  in  which  the  diagonal  Fig.  82, 
ID  represents  the  force  exerted  upon  the  body  of  the  machine,  ' ' 
IJl  being  considered  as  representing  the  tension  of  the  rope    O  p 

or  OG.     Now  since  IT^  10,  are  tangents,  and 

IB  =  JA, 

it  will  be  seen  that  ID  produced  would  pass  through  the  centre  F 
of  the  pulley.  Therefore,  if  the  body  of  the  pulley  is  not  firmly 
fixed,  JD  cannot  be  destroyed,  except  the  obstacle,  whatever  it  be, 
which  is  to  prevent  the  motion  of  the  body  of  the  pulley,  is  situated 
in  some  point  of  the  line  IF,  extending  from  tlie  centre  jp  to  the 
point  of  meeting  of  the  two  ropes.  Thus,  if  the  pulley  is  destined 
to  turn  in  a  block  FG,  fixed  to  some  point  G  without,  and  ad- 
mitting of  a  motion  about  G,  an  equilibrium  will  not  take  place  ex- 
cept when  the  block  has  the  direction  FI.  Fig.  83, 

In  like  manner,  if  the  body  of  the  pulley,  being  embraced  by 
a  rope  fixed  to  the  point  G,  is  movable,  there  will  not  be  an  equi- 
librium, except  the  effort  applied  at  the  centre  F,  or  to  the  fixed  Fig.  8L 
block  at  this  centre,  is  exerted  in  such  a  direction  as  to  bisect  the 
angle  formed  by  the  two  parts  of  the  rope  OG,  T q,  and  is  at  the 
same  time  to  the  tension  of  OG,  T q, 

::  ID  :  lA   :  IB. 

177.  It  is  now  easy  to  find  the  ratio  of  the  tension  of  each  part 
of  the  rope  that  passes  round  the  pulley,  to  the  force  exerted  upon 
the  body  of  the  pulley,  and  consequently  to  the  force  of  which  the 
movable  pulley  is  capable.  The  tension  of  each  part  of  the  rope 
being  represented  by  lA  or  its  equal  IB,  the  effort  which  is  exerted  Fig  84. 
upon  the  body  of  the  pulley,  will  be  expressed  by  ID.  But  in  the 
triangle  JAD. 


104  Statics. 

M  :  ID   :  :  sin  IDA  :  sin  MD, 
sin  FI  q  :  sin  OAD, 
sin  Fl  q  :  sin  Gl  q. 

We  may  say,  therefore,  universally,  that  when  there  is  an  cqnilib^ 
rium  by  means  of  the  simple  indley,  fixed  or  movable  ;  (1.)  The 
tensions  of  the  two  parts  of  the  rope  which  passes  round  the  pulley, 
or  the  poivers  applied  to  them,  are  equal ;  (2.)  That  each  of  these 
poicers  is  to  the  force  exerted  at  the  centre  of  the  pulley,  as  the  sine 
of  half  the  angle  formed  by  the  two  parts  of  the  rope  in  question, 
is  to  the  sine  of  the  ivhole  of  this  angle. 

Fig,  82,  Thus  in  the  fixed  pulley,  there  is  no  other  advantage  gained  by 
the  agent  q,  except  that  of  being  able  to  change  at  pleasure  the 
direction    in    which    the    action   shall  be   employed.      But  in    the 

Fig.  84,  movable  pulley,  there  is  possessed  by  the  agent  q,  the  double  ad- 
'  ■  vantage  of  a  change  of  direction  and  an  augmentation  of  the  effect 
of  the  action.  But  it  is  to  be  remarked,  that  according  as  the  di- 
rection is  changed,  the  force  exerted  upon  the  centre  varies,  so  that 
there  is  a  direction  in  which  the  effect  produced  by  a  given  power 
is  the  greatest  possible  ;  and  this  is  when  the  two  parts  of  the  rope 
GO,  T  q,  are  parallel,  as  will  be  readily  perceived. 

Fig.  84.        178.   If  we  draw  the  radii  OF,  FT,  and   the   chord   OT,   the 

triangle  OFT,  having  its  sides  perpendicular  respectively  to  those 

Geom.    of  the  triande  BID,  will  be  similar  to  BID  ;  whence 
209.  ^  ' 

IB  :  ID  ::  FT  :   OT, 

or 

q  :    r     '.  :  FT  :   OT; 

that  is,  the  tension  of  either  part  of  the  rope  is  to  the  force  exerted 
against  the  centre  F,  as  the  radius  of  the  pulley  is  to  the  chord  of 
the  arc  embraced  by  the  rope. 

Now  it  is  evident  that  this  last  ratio  is  the  greatest  possible 
when  the  two  parts  of  the  rope  are  parallel  ;  hence  in  the  mov- 
able pulley  the  power  is  the  least  possible,  or  is  exerted  to  the  great- 
est advantage,  when  the  two  parts  of  the  rope  are  parallel  ;  and  it 
is  then  half  of  the  force  exerted  against  the  centre  of  the  pulley. 
This  second  kind  of  pulley  is  made  use  of  in  tightening  the  sails 


Pulley.  105 

of  a  vessel,  by  attaching  it  to  one  of  the  corners  as  represented  in 
figure  86. 

179.  If,  therefore,  the  weight  p  is  sustained  by  the  power  ^r,  Fig.  87. 
by  means  of  several  movable  pulleys,  embraced  each  by  a  rope, 
one  extremity  of  which  is  attached  to  a  fixed  point,  and  the  other 
to  the  block  of  a  pulley,  the  ratio  of  the  power  to  the  weight  will 
be  that  of  the  product  of  the  radii  of  all  the  movable  pulleys  to 
the  product  of  the  chords  of  the  arcs  embraced  by  the  ropes» 

Indeed  if  we  call  cr,  q,  the  forces  exerted  at  the  centres  of  the 
pulleys  JV,  M,  which  are  at  the  same  time  the  tensions  respective- 
ly of  the  two  ropes  attached  to  the  centres  of  JV  and  M ;  r,  r', 
r'',  being  the  radii,  and  c,  c',  c",  the  chords  of  the  arcs  embraced 
by  the  ropes  in  the  several  pulleys  JV*,  M,  L,  we  shall  have  178. 


?  '- 

cr  : 

:  R 

:  c. 

cr  : 

;   9 

:  :  R' 

:  c', 

?   • 

P  ' 

:  :  R^' 

:  c"; 

whence, 

by  taking  the 

prodi 

net  of  the 

corresponding 

terms. 

?^§ 

1   :  cr 

gp 

:  :  rr'r"  : 

c  cV, 

or 

Alg.226. 


q         :     p      :  :   rr'r''  :  c  &c"  j 
that  is,  when  the  cords  are  parallel,  which  gives 

c  =  2r,         c'  =  2  r',         c"  =  2  R'', 

q  :  p  :  :  rr'r"  :  2  r  X  2  r'  X  2  r", 

:  :  1  :2  X  2  X  2; 

in  other  words,  the  power  is  to  the  weight  as  unity  to  the  num- 
ber 2  raised  to  the  power  denoted  by  the  number  of  movable 
pulleys.  With  three  pulleys,  for  example,  the  power  would  sustain 
a  force  eight  times  as  great. 

180.  But  this  arrangement  of  pulleys  is  not  the  most  conven- 
ient. It  is  more  common  to  employ  one  of  the  forms  represented  in 
figures  88,  89,  90,  91,  92,  in  which  all  the  pulleys,  both  fixed  and 
movable,  are  embraced  by  the  same  rope.  Moreover  all  the  fixed 
pulleys  are  attached  to  one  block,  and  all  the  nwvable  pulleys  to 
another.  Sometimes  the  centres  are  distributed  upon  different 
points  of  the  same  block  as  in  figures  88,  89,  90,  91.  Some- 
Mech.  14 


106  Statics. 

times  they  are  united  upon  the  same  axis  as  in  figures  92,  93. 
The  latter  arrangement  has  the  advantage  of  being  more  compact ; 
but  when  a  large  number  of  pulleys  are  thus  disposed  in  the  same 
block,  the  power  being  applied  on  one  side  instead  of  being  directed 
through  the  middle,  the  system  is  drawn  awry,  and  part  of  tlie  force 
employed  is  lost  by  the  oblique  manner  in  which  it  is  exerted. 
This  inconvenience  does  not  belong  to  the  pulleys  represented  in 
figures  88,  89,  90,  91  ;  and  in  that  represented  by  figure  94,  the 
peculiar  advantages  of  the  two  systems  are  united.  Here  two  sets 
of  pulleys  having  a  common  axis  are  attached  to  the  movable  block, 
and  two  to  the  fixed  block,  the  inner  set  in  each  case  being  of  a 
less  diameter  than  the  outer,  so  as  to  allow  a  fi-ee  motion  to  the 
rope.  Then  the  rope  commencing  at  the  middle  of  the  upper 
block,  after  being  made  to  pass  over  all  the  pulleys,  will  terminate 
also  in  the  middle.     This  arrangement  was  invented  by  Smeaton. 

181.  But  whatever  difference  there  may  be  in  this  respect  in 
the  particular  disposition  of  the  pulleys,  the  ratio  of  the  power  to 
the  weight  may  always  be  found  by  the  following  rule.  The  power 
is  to  the  weight  as  radius,  or  sine  of  90,  is  to  the  sum  of  the  sines  of 
the  angles  made  by  the  several  ropes  (meeting  at  the  movable 
pulley)  with  the  horizon. 

Fig.  88,  Indeed,  if  upon  each  of  the  ropes  we  take  the  equal  parts  IM, 
JVP,  Slc,  to  represent  the  tension,  and  upon  each  of  these  lines,  as 
a  diagonal,  we  form  a  parallelogram,  having  one  pair  of  its  opposite 
sides  vertical,  and  the  other  pair  horizontal ;  instead  of  considering 
the  weight  p  as  sustained  by  the  immediate  tension  of  the  ropes,  we 
may  regard  it  as  supported  by  the  horizontal  forces  IK,  NO,  &£c.,  and 
40.  the  vertical  forces  IL,  JV(^,  &c.  Now  the  first,  being  perpendicular 
to  the  action  of  the  weight,  contribute  nothing  to  counterbalance  this 
action  ;  and  in  the  case  of  an  equilibrium  these  horizontal  forces 
mutually  destroy  each  other.  The  weight  p,  therefore,  is  wholly 
sustained  by  the  resultant,  that  is,  by  the  sum  of  the  vertical  forces 
IL,  JV(^,  ^c. ;  and,  the  ropes  being  all  equally  stretched,  it  is  evi- 
dent that  q  IS  to  p  as  the  tension  of  one  of  these  ropes  is  to  the 
entire  sum  of  the  vertical  forces.  But  in  the  right-angled  triangles 
IML,  JVqP,  &c.,  we  have 

IM  :  IL  ::1   :  sin  IML ;  JVP  or  IM  :  JVq  ::  I  :  sin  JVPQ; 


N  Pulley.  107 

the  same  may  be  said  of  the  other  ropes  ;  whence 

IL  =  IM  sin  IML ;    JVq  =  IM  sin  JVPQ ; 

accordingly, 

q  :  p  ::  IM  :  IMs\n  IML  +  IM  sin  JYPq  +  &;c., 
:  :     1      :  sin  IML  +         sin  ATQ  +  ^c. 

If  the  ropes  are  parallel  and  consequently  vertical,  the  angles 
IML,  JVPq,  4^0.,  will  be  right  angles,  and  their  sines  will  be  each 
equal  to  radius,  or   1.     Therefore  the  power  in  this  case  will  be  to 
the  weight  as  1  is  to  the  sum  of  so  many  units   as    there  are  ropes 
meeting  at  the  movable   pulley.     Hence  it  will  be  seen,  that  if  one 
of  the  extremities  of  ike  rope  is  attached  to  the  fixed  pulley ,  the  poiver  Y\g.  as 
will  be  to   the  weight  as  unity  is  to  double  the  number  of  movable  ^' 
pulleys ;  and  if  the  extremity  of  the  rope  is  attached  to  the  movable  ^'S'  89, 
pulley,  the  power  will  be  to  the  weight  as  unity  is  to  double  the  num- 
ber of  movable  pulleys,  plus  1. 

182.  The  general  proposition  above  demonstrated  holds  true, 
whether  the  ropes  are  in  the  same  plane  or  not ;  and  if  the  obstacle 
to  be  overcome  be  not  a  weight,  that  is,  if  the  direction  of  the  whole 
power  of  the  pulley  be  not  vertical,  we  have  only  to  substitute  for 
the  angles  which  the  ropes  are  supposed  to  make  with  the  horizon, 
those  which  they  would  make  with  a  plane  perpendicular  to  the 
whole  action  of  the  pulley.  In  figure  93,  for  example,  the  power 
q  is  to  the  force  exerted  at  G,  as  radius  is  to  the  sum  of  the  sines  of 
the  angles  made  by  the  several  ropes  (meeting  in  CF)  with  a  plane 
perpendicular  to  FG. 

183.  If  several  sets  of  pulleys  are  employed,  it  will  be  easy 
after  what  has  been  said  to  assign  the  ratio  of  the  power  to  the 
weight.  In  figure  93,  for  example,  the  ropes  being  supposed  paral- 
lel, the  power  q  will  be  to  the  force  exerted  in  the  direction  CB,  *81. 
as  1  is  to  5.  Now  this  last  force  performs  the  office  of  a  power  with 
respect  to  the  system  of  pulleys  BA,  and  accordingly  is  to  the 
weight  p,   as   1  to  4.     Therefore  the  power  q  is  to  the  weight  p, 

as  1   X  1  is  to  4  X  5,  that  is,  as  1  to  20. 

184.  In  all  that  precedes,  we  have  supposed  the  system  of  pul- 
leys destitute  of  gravity  and  friction,  and  the  ropes  perfectly  flexible. 
We  shall  see  hereafter  what  allowance    is  to  be  made  for  friction 


108  Statics. 

and  the  stiffness  of  the  ropes.     With  respect  to  the  gravity  of  the 

parts  of  the  system  whicii  the  power  has  to  sustain,  allowance  is 

made  for  it,  in  the  case  of  an  equilibrium  by  adding  it  to  the  weight, 

Fig.  90,  when  its  action  coincides  with  that  of  the  weisrht.     But  if,  as  in  fig- 
91  o  '  >j 

ure  93,  the  gravity  of  the  system  CF  is  not  exerted  in   the  same 

direction  with  the  power  q,  BC,  instead  of  being  in  the  same  direc- 
tion with  the  power  q,  is  in  the  direction  of  the  resultant  of  the 
gravity  of  the  system,  and  the  force  exerted  independently  of 
gravity. 


Of  the   Wheel  and  Axle. 

185.  The   wheel  and  axle  consists  in   general   of  a  grooved 
Fig.  96,  wheel,  and  a  cylinder  passing  perpendicularly  through  the  centre 

of  the  wheel,  and  resting  at  its  extremhies  upon  two  fixed  sup- 
ports P,  F.  A  power  q,  applied  in  the  direction  of  a  tangent  to 
the  circumference  of  the  wheel,  turns  this  wheel,  together  with  the 
cylinder,  which  being  firmly  fixed  to  it,  takes  up  successively  the 
different  parts  of  the  cord  Dp  and  with  it  the  weight  p,  which  ^it 
is  proposed  to  elevate  or  draw  toward  the  cylinder. 

Fig.  96,        Somefimes  instead  of  a  wheel,  bars  E,  E,  in  the  form  of  radii 
98, 99.    are  the  points  at  which  the  power  is  applied,  and  by  which  the  same 
effect  is  produced.     At  other  times  the  extremities  of  the   cylinder 
Fig.  97.  ^^^  provided  with  winches  q,  q,  at  which  the  moving  force  is  ex- 
erted. 

When  the  axis  of  the  cylinder  is  vertical  the  machine  is  called 
lOo!  ^^'  ^  ^o.'P^io,'""  It  is  in  this  position  that  it  is  used  on  board  of  vessels, 
with  this  difference  in  the  construction,  however,  that  the  figure  of 
the  axis  is  made  conical  instead  of  being  cylindrical,  that  it  may  be 
worked  more  easily  when  the  rope,  having  reached  the  lowest  point, 
in  turning  to  retrace  its  course  would  tend  to  check  the  motion. 

186.  But  however  the  machine  is  placed,  it  will  be  seen  that 
the  action  of  the  power  and  that  of  the  weight  which  it  is  proposed 
to  raise,  are  not  exerted  in  the  same  plane,  but  in  planes  that  are 
parallel  or  nearly  so.  The  power  produces  two  effects,  one  of 
which  is  exerted  against  the  weight,  and  the  other  against  the  sup- 
ports. In  case  of  an  equilibrium,  these  effects  may  be  determined 
in  the  following  manner. 


Wheel  and  Axle.  109 

The  essential  parts  of  the  machine  are  represented  in  figure 
101,  where  AMJV  is  the  plane  of  the  wheel,  FF  the  axis  of  the 
cylinder,  and  BDL  a  section  of  the  cylinder,  parallel  to  AAIJV, 
and  passing  through  the  cord  D  p. 

Having  drawn  the  radius  EA  to  the  point  A,  where  the  power 
q  acts  upon  the  wheel,  suppose  a  plane  FEA  passing  through 
FF,  and  EA,  and  meeting  BDL  in  IB ;  IB  will  be  parallel  to 
EA.  Join  AB,  and  through  this  line  and  the  direction  A  q  oi 
the  power,  imagine  a  plane  q  AG  to  pass  meeting  the  axis  FF  in 
some  point  G.  Lastly  through  B  and  G  draw  B  xs  and  G  r 
parallel  each  to  A  q. 

This  being  supposed,  the  force  q  may  be  decomposed  into  two 
other  forces  sr,  r,  directed  according  to  jB  cr,  G  r ;  and  as  this  53. 
last  passes  through  the  axis  of  the  cylinder,  it  can  have  no  effect  in 
turning  the  machine  about  this  axis,  and  consequently  can  contrib- 
ute nothing  toward  the  support  of  the  weight  p.  It  will  therefore 
be  expended  against  the  supports  jP,  F.  There  will  accordingly  be 
only  the  force  nr  by  which  an  equilibrium  with  the  weighty  is  to  be 
effected.  Now  (1.)  This  force  is  directed  in  the  same  plane 
BDL  in  which  the  action  of  the  weight  is  exerted.  (2.)  The  two 
lines  B  ct,  B1,  being  parallel  respectively  to  the  two  A  q,  AE, 
which  are  at  right  angles  to  each  other,  B  ru  \s  perpendicular  to  BI,  Geom. 
and  consequently  a  tangent  to  the  circumference  BDL.  We  may 
therefore  consider  BID  as  an  angular  lever,  of  which  the  fulcrum 
is  at  1 ;  and  since  the  distances  of  the  direcuons  of  the  two  powers 
nr,  p,  from  the  fulcrum  are  equal,  these  two  powers  must  be  equal ; 
we  have  accordingly  ^  =■  p.  Let  us  now  see  what  is  the  ratio  of 
C7  to  q. 

According  to  what  has  been  laid  down,  we  have 

q  :  ^  ::  BG  :  AG;  52. 

but  the  similar  triangles  GBI,  GAE,  give 

BG  :  AG  :  :  BI  :  AE; 


whence 

or,  since  w  :=  p, 


q  :  xa       ::  BI  :  AE; 
q  :  p       ::  BI  :  AE; 


1 10  Statics. 

that  is,  in  the  wheel  and  axle  the  power  is  to  the  weight,  as  the 
radiiLs  of  the  cylinder  to  the  radius  of  the  wheel. 

Fig.102.        1 87.  If  the  weight  p  be  attached  at  some  point  B  in   the   plane 
of  the   wheel,  in  such  a  manner,  that  the  perpendicular  IB  upon  its 
direction  shall  be   equal  to  the  radius  of  the  cylinder,  we  may  con- 
sider AlB  as  an  angular  lever  the  fulcrum  of  which  is  at  the  centre 
159.     /.  2|^{]  \^  order  to  an  equilibrium,  we  must  have 

q  :  p  :  :  BI  :  Al, 

that  is,  the  ratio  between  the  power  and  the  weight  would  be  the 
same  as  the  above.  Therefore  the  action  of  the  power  is  trans- 
mitted to  the  weight  by  means  of  the  wheel  and  axle,  in  the  same 
manner  as  if  the  power  and  the  weight  were  in  the  same  plane. 

188.  It  is  not  the  same,  however,  with  respect  to  the  force 
exerted   against  the   supports.     This  varies  according  to  the  dis- 

Fig.ioi.tance  of  the  plane  BDL  from  the  plane  of  the  wheel.  In  order 
to  determine  what  it  is,  we  decompose  the  power  q,  considered 
as  applied  at  E  parallel  to  A  q,  into  two  forces  parallel  to  A  q, 
55.  and  passing  through  F  and  jP.  We  decompose  likewise  the  pow- 
er p,  considered  as  applied  at  i,  into  two  forces  parallel  to  p  D, 
and  passing  through  jP  and  F.  By  this  means  each  support  will 
be  urged  by  two  forces,  the  magnitude  and  directions  of  which 
will  be  known.  It  will  be  easy,  therefore,  to  reduce  these  forces, 
in  the  case  of  each  support,  to  a  single  one  of  a  known  magnitude 
and  direction. 

This  method  of  finding  the  forces  exerted  against  the  two  sup- 
ports, is  founded  upon  the  fact,  that  the  two  forces  w  and  p  reduce 
themselves  to  one  which  acts  at  /.  If  we  conceive  this  decom- 
posed into  two  forces  parallel  to  the  direction  sr  and  p,  and  applied 
in  1,  they  will  have  simply  the  values  of  to  and  p.  Accordingly, 
(1.)  We  may  regard  p  as  applied  at  i;  (2.)  The  force  sr,  consid- 
ered as  applied  at  /,  and  the  force  r  applied  at  G,  cannot  but  have 
for  a  resultant  the  force  q,  by  which  they  are  produced,  as  we  have 
52.    seen  above  ;  moreover  this  resultant  passes  through  E,  since 

Gl  :   GE  ::   GB  :   GA  ::  q  :  tir. 

189.  If  the  power,  instead  of  being  applied  in  the  direction  of 
100,  ^^'  ^  tangent  to  the  wheel,  acted  by  means  of  the  arms  EE,  and  at 


Wheel  and  Axle.  Ill 

right  angles  to  their  length,  the  ratio  of  the  power  to  the  weight 
would  always  be  found  to  be  the  same  as  above  stated,  by  substitut- 
ing for  radius  of  the  wheel  the  words  length  of  the  arm ;  this  length 
being  reckoned  from  the  axis  of  the  cylinder.  But  if  the  power 
acted  in  a  direction  not  perpendicular  to  the  arm  IE,  instead  of  the  Fig.  99. 
length  of  the  arm  we  should  take  that  of  the  perpendicular  IR  let 
fall  upon  the  direction  of  the  power ;  so  that  in  this  case  the  power 
will  be  to  the  weight  as  the  radius  of  the  cylinder  to  the  perpen- 
dicular IR. 

190.  Since  q  :  p  :  :  BI :  AE,  we  have  g  X  AE  =p  X  BI;rig.ioi. 
that  is,  the  moment  of  the   power  is  equal  to   the   moment  of  the 
weight,  these  moments  being  taken   with  respect  to  the  axis  FF, 

If,  therefore,  several  powers  are  employed  at  the  same  time,  applied 
to  different  arms,  the  sum  of  the  moments  of  these  powers  must 
be  equal  to  the  moment  of  the  weight. 

191.  If  the  cord  which  supports  the  weight  or  which  transmits 
the  action  of  the  power  to  the  weight,  were  wound  round  a  conical 
surface,  or  a  surface  of  a  variable  diameter,  instead  of  that  of  a 
cylinder,  the  ratio  of  the  power  to  the  weight,  would  also  vary 
continually ;  and  reciprocally,  if  the  power,  whose  action  is  to  be 
communicated  through  the  medium  of  such  a  juachine  as  that 
under  consideration,  varies  continually,  and  is  intended,  notwith- 
standing, to  produce  the  same  effect,  we  arrive  at  the  end  proposed, 
by  causing  the  action  to  be  applied  successively  to  radii  that  increase 
in  length  according  as  the  power  diminishes.  We  have  an  exam- 
ple of  this  adaptation  of  the  machine  to  a  varying  power  in  watches 
and  chronometers,  in  which  the  moving  or  maintaining  power  is  a 
spring  fixed  at  one  of  its  extremities  to  the  axis  or  arbor  of  a  bar- 
rel Z,  and  which,  after  several  revolutions  or  coils,  is  attached  to  the  Fig  103. 
interior  of  this  barrel.     A  chain  with    one  of  its  extremities  fixed 

to  the  convex  surface  of  the  barrel  is  wound  round  the  conical  axis 
or  fusee  Y,  to  which  the  other  extremity  of  the  chain  is  attached. 
As  the  spring  uncoils,  the  barjel  turns,  and,  drawing  the  chain, 
causes  the  fusee  to  turn  ;  but  since  the  force  of  the  spring  dimin- 
ishes as  it  uncoils,  a  compensation  is  made  for  this  reduction  of  the 
power,  by  giving  a  greater  diameter  to  those  parts  of  the  fusee  on 
which  the  last  coils  of  the  spring  are  exerted.  By  this  contri- 
vance, the  machinery  receives  nearly  equal  impulses  in  equal 
times. 


y 


112  Statics. 


192.  It  would  seem,  therefore,  by  having  regard  only  to  an 
equilibrium,  that  we  might  diminish  at  pleasure  the  ratio  of  the 
power  to  the  weight,  and  make  a  force,  however  small,  counter- 
balance one  however  large,  by  means  of  the  wheel  and  axle,  and 
such  machines  as  depend  upon  the  same  principle.  But  if  we  take 
into  consideration  their  motion,  and  have  respect  also,  as  we  must, 
to  the  nature  of  the  agents  to  be  employed,  we  cannot  augment 
the  effect  at  pleasure.  The  ratio  of  the  radius  of  the  cylinder 
to  that  of  the  wheel,  is  not  arbitrary.  It  requires  a  particular  adap- 
tation to  the  purpose  proposed,  in  order  to  produce  the  greatest 
possible  effect. 

Fig.  99.  Suppose,  for  example,  that  the  agent  applied  to  the  arm  E, 
tends  to  move  with  a  velocity  u,  and  that  the  force  of  which  it 
is  capable,  is  m  u,  that  is,  equal  to  a  known  mass  m  urged  with 
a  velocity  u.  Let  v  be  the  velocity  with  which  the  point  E 
would  be  moved  in  virtue  of  the  resistance  of  p  ;  then,  if  we  call 
D  the  perpendicular  distance  of  E  from  the  axis,  and  8  that  of  p 
from  the  axis,  we  shall  obtain  the  velocity  that  p  would  have,  by 
the  proportion, 

since  it  is  evident,  that  the  point  E  and  the  point  where  the  cord 
touches  the  cylinder,  would  have  velocities  proportional  to  their  dis- 
tances from  the  axis. 

We   must   suppose,    therefore,   that  at   the   instant  when    the 

power  comes  to  exert   itself,  the  velocity  u   is  composed  of  the 

133.     velocity  v,  which    actually    takes    place,    and    the    velocity   u  —  v, 

which  is  destroyed ;  and  that  at  the  same    instant  the  weight  p    has 

the  velocity  -^,  which  actually  takes  place,   and    the    velocity    -jy 

in  the  contrary  direction,  which  is  destroyed ;  that  is,  the  moving 
force  m  {u  —  v)  must  be  in  equilibrium  with  the  mass  p,  urged 

with  the  force  —jj- .        Accordingly, 

7)  8^  V 

m   {u  —v)  X  D  =  f-^  ; 

whence 

m  u  D^ 

^~  m  D^-\-p  d^  ' 


Wheel  and  Axle.  113 


and  the  velocity  of  p,  namely  -jj-  will  be 

m  u  8  D 


Therefore,  in  order  to  know  what  ratio  there  must  be  between  D 
and  J,  in  order  that  2?  may  have  the  greatest  velocity   possible,   it   iscal.  48. 
necessary  to  put  equal  to  zero    the    differential   of  this  expression, 
taken  by  regarding  8  only  as  variable  ;  thus 

m  u  D  d  8  {m  D^  -]-  p  8^)  —m  u  B  8  X  2p  8  d  8  =  0, 

or 

muDd8{mD^-l-p8'^)—'muD8  X  22^  8d  8  =  0, 
whence 

mD^ -\-p8^  —  2p8^      or     mIP—p8^  =  0, 
which  gives 


\    P  \P 


If,  for  example,  the  weight  p  be  1000001b.,  and  the  mass  m  or  mov- 
ing force  be  equivalent  to  a  weight  of  lOlb.,  we  shall  have 


^  =  ^J.iL.=  i>x 


100000  —  -^  ^  Tssy 


that  is,  the  radius  of  the  cylinder  must  be  a  hundredth    part   of  the 
arm  IE,  in  order  that  the  effect  may  be  the  greatest  possible. 

193.  There  are  many  machines  which  are  referrible,  either 
wholly  or  in  part,  to  the  wheel  and  axle,  and  consequently  to  the 
lever ;  such  as  rack-work,  machinery  in  which  wheels  are  con- 
nected by  bands,  tooth  and  pinion  work,  and  instruments  intended 
for  drilling,  boring,  and  screwing,  although  these  last  operations 
often  depend  in  part  upon  another  machine  that  remains  to  beFig.105. 
described,  namely,  the  inclined  plane.  In  rack-work,  the  axis 
FE,  having  a  ivinch  FR  q,  carries  a  pinion  the  teeth  or  leaves 
of  which  act  upon  the  toothed  bar  AB.  The  leaves  of  the  pin- 
ion, in  turning,  raise  the  bar  AB  with  a  force  which  is  to  the 
force  q  applied  to  the  winch,  as  the  radius  of  the  winch  is  to  that 
of  the  pinion  j  and  as  the  radius  of  the  pinion  is  for  the  most  part 
small  compared  with  that  of  the  winch,  by  the  aid  of  such  a 
Mech.  15 


114  Statics. 

machine   we  are  able  Lo  raise   a   veiy   considerable  weight  with  a 
moderate  force.  , 

194.  Toothed  wheels  serve  several  purposes.  Sometimes  wc 
employ  thein  to  augment  a  force,  at  others  to  increase  a  velocity, 
often  to  change  the  direction  of  a  motion,  and  still  more  frequently 
to  adapt  motion  to  certain  periods  of  time,  or  to  render  sensible 
certain  motions  or  spaces,  that  the  eye  cannot  distinguish. 

Fig.  106.  Several  toothed  wheels  TV,  X,  Y,  Z,  being  connected  together 
by  the  pinions  iv,  x,  y,  z,  it  is  proposed  to  find  the  ratio  of  the 
power  (J,  applied  to  the  first  wheel,"  to  the  weight  or  effort  p, 
sustained  by  the  last  pinion.  Let  D,  D',  D",  D"\  be  the  greater 
distances  or  radii  of  the  wheels,  ^,  6',  8",  8'",  the  less  distances 
or  radii  of  the  pinions.  We  shall  consider  the  effort  made  by  the 
leaf  of  any  one  of  the  pinions  upon  the  tooth  of  the  neighbouring 
wheel,  as  a  power  applied  to  this  last ;  then  E,  E',  E",  being 
186.    these  efforts,  we  shall  have 

q    :  E  :  :  S    :  D,  E     :  E'  :.  8'     :  D\ 

E'  :  E"  :  d''  :  -D'^        E"  :  p    :  :  8'"  :  D"  ; 
whence,  by  taking  the  products  of  the  corresponding  terms, 
q  :  p  :  :  8  '  8'  •  8"  '  8'"  :  D  •  D'  -  D"  '  B'" ; 

that  is,  the  power  is  to  the  weight  as  the  product  of  the  radii  of  all 
the  pinions  to  the  product  of  the  radii  of  all  the  wheels. 

If,  for  example,  the  radius  of  each  pinion  is  one  tenth  of  that 
of  the  corresponding  wheel,  we  should  have 

gr  :  p  :  :  1   :  10   X    10   X    10   X    10, 
1   :  10000; 

that  is,  a  power  of  one  pound  would  counterbalance  a  weight  of 
10000  pounds. 

What  is  gained,  however,  in  point  of  force  by  the  use  of 
wheels  and  pinions,  is  lost  in  respect  to  velocity.  Indeed  while 
the  wheel  W  turns  once,  the  pinion  w,  turning  in  the  same  time, 
Fig.  106.  causes  to  pass  only  as  many  teeth  of  the  wheel  X,  as  it  has  leaves 
in  its  own  circumference,  so  that  if  the  wheel  X  has  48  teeth,  and 
the  pinion  w  six  leaves,  the  wheel  X  would  make  only  /g  or  | 
part  of  a  revolution,  while  W  turns  once  round ;  it  will  hence  be 


Wheel  and  Axle,  115 

seen  that  the  wheel  X  goes  just  so  much  slower  than  W,  Y  so 
much  slower  than  X,  and  so  on. 

195.  From  what  is  above  said,  it  will  be  perceived  how,  by 

means  of  toothed  wheels,  the  velocity  may  be  augmented   in   any 

given  ratio.     Let  there  be,   for  example,  the  toothed  wheel    TV,  Fig.  107. 

acting  upon  the  pinion   iv ;  it  is  clear,  that  during  one  revolution 

of  fV,  the  pinion  lo  will  turn  as  many  times  as  the  number  of  leaves 

in  the  pinion  is  contained  in  the   number  of  teeth  of  the  wheel ; 

N 
that  is,  during  one  revolution  of  the  wheel,  the  pinion  will  turn  — 

times,  JV  denoting  the  number  of  teeth  in  the  wheel,  and  v  the 
number  of  leaves  in  the  pinion. 

If  therefore  the  axis  of  the  pinion  iv  carries  a  wheel,  which 

acts   also  on  a  pinion  x,  we  shall  see    that  during  one  revolution 

N' 
of  the  wheel  X,  or  of  the  pinion   iv,  the   pinion  x  will   turn  — r 

times,  JV  denoting  the  number  of  teeth  in  the   wheel  X,  and  v'  the 

number  of  leaves  in  the  pinion  x.     Therefore  while  the  wheel  X 

N 
makes  a  number  of  turns   expressed  by  — ,  that  is,   during   one 

revolution  of  the   wheel    PV,  the  pinion  x  revolves   a  number  of 

N'        N  N'N 

times  expressed  by  —r  X  —      or     — 7—.     And    by   reasoning  in 

this  manner  for  a  greater  number  of  wheels  and  pinions,  it  will  be 
perceived  that  the  number  of  times  that  the  last  pinion  turns,  during 
one  revolution  of  the  first  wheel,  is  expressed  by  a  fraction  having 
for  its  numerator  the  product  of  the  number  of  teeth  in  the  several 
wheels,  and  for  a  denominator  the  product  of  the  number  of  leaves 
in  the  several  pinions. 

When  it  is  asked,  therefore,  what  must  be  the  number  of  teeth 
and  leaves  for  a  proposed  number  of  wheels  and  pinions,  in  order 
that  the  velocity  of  the  last  piece  shall  be  to  that  of  the  first  in  a 
given  ratio,  the  question  is  indeterminate,  that  is,  one  wliicli  admits  of 
several  answers.  Two  examples  will  sullice  to  show  how  we 
ought  to  proceed  in  questions  of  this  kind. 

We  will  suppose  that  it  is  required  to  find  hov/  many  teeth  must 
be  given  to  the  two  wheels  fV  and  X,  and  how  many  leaves  to  the 
pinions  w  and  x,  in  order  that  the  pinion  x  may  make  60  revolu- 
tions while  the  wheel  W  makes  one.     We  shall  have 


116  Statics. 

NN' 

^  =  50. 

We  know  in  this  case  only  the  quotient  obtained  by  dividing  NN' 
by  V  v' ',  we  do  not  know  either  the  dividend  or  the  divisor. 
Let  us  take,  therefore,  arbitrarily  for  the  divisor  v  v'  a  number 
composed  of  two  factors  which  shall  be  neither  too  small  nor  too 
great  for  the  number  of  leaves  to  be  allowed  to  the  pinions.  Sup- 
pose, for  example,  vj/'  =  7x8  =  56,  »'  being  7,  and  v'  8.     We 

NIV' 
shall  then  have  ^  =  50,  or  NJV'  =  50  X  56.      Now  50  and 
oo 

56  not  exceeding  the  number  of  teeth  that  can  be  given  to  the 

wheels  W  and  X,  I  will  suppose  N  to  have  50  ;  and  consequently 

those  of  N  will  be   56.     If  these  two  factors,  or  one  of   them, 

should  happen  to  be  too  great,  I  should  decompose  them   into  their 

prime   factors,   and  see  if  from  the  combination   of  these   factors 

there  would  not  result  two  smaller  factors  j  or  another  number  might 

be  taken  for  v  v'. 

Suppose,  for  a  second  example,  that  it  is  proposed  to  find  the 
number  of  teeth  and  leaves  to  be  given  to  three  wheels  and  their 
pinions,  in  order  that  while  the  last  pinion  turns  once  in  twelve 
hours,  the  first  wheel  shall  require  a  year  to  make  one  revolution. 

The    common   year  consisting   of    365,25    X    24    X    60   or 

525949  minutes,  and   12  hours  being  equal  to   12  X  60  or  720 

minutes,  it  is  evident  that  during  one  revolution  of  the  first  wheel 

the  last  pinion  will  make  a  number  of  revolutions  expressed  by 

JVN'N" 
52  5  949     ^g  have,  therefore,  — ; — ;;  =    'VoV^*      Let  us  take 

NN'N" 
arbitrarily  v  =  7,  v '  =  8 ;  and  we  shall  have        ^   „  = 


S  2_S_9_4  9 


or  NNN'  =  ^ViV  X  7  X  8  v"  = 


Xi 

3681643  v" 
90 


Of  the  Inclined  Plane. 

Fig.  108.  196.  If  a  body  p  of  any  figure  whatever,  touching  a  plane 
XZ  in  any  point  C,  is  urged  by  a  single  force,  it  can  remain  at 
rest  on  this  plane  only  when  the  direction  of  this  force  is  perpen- 
dicular to  the  plane,  and  is  such  at  the  same  time  as  to  pass  through 


Inclined  Plane.  117 

the  point  C.  The  necessity  of  the  first  condition  is  evident.  As 
to  the  second,  it  will  be  seen,  with  a  moment's  attention,  that  this  is  38. 
not  the  less  necessary  ;  since,  if  the  direction  AD  of  the  body  j/, 
for  example,  although  perpendicular  to  the  plane,  does  not  pass 
through  the  point  of  contact  C%  the  resistance  of  the  plane,  which 
cannot  be  exerted  except  according  to  the  perpendicular  at  C, 
would  not  be  directly  opposed  to  the  force  AD,  and  conse- 
quently would  not  destroy  it,  even  when  it  is  supposed  equal  to  this  137. 
force. 

197.  If  the  body,  instead  of   touching  the  plane  only  in  one 
point,  touches  it  in  several   points,   it  is  not  indispensable   that  theFig.i09, 
single  force  AD,  which  acts  upon  it,  should  pass  through   any  one 

of  these  points ;  but  it  is  necessary  that  it  should  be  perpendic- 
ular to  the  plane,  and  that  it  should  be  capable  of  being  decom- 
posed into  as  many  forces  perpendicular  to  the  plane,  as  there 
are  points  which  rest  upon  it,  and  that  they  should  be  such  as  to 
pass  through  these  points.  Thus  if  the  body  p,  for  example,  were 
in  contact  with  the  plane  at  the  points  C,  C',  and  the  force  -4jDFig.l09. 
were  not  in  the  plane  which  passes  through  the  two  perpendicu- 
lars raised  at  the  points  C,  C',  an  equilibrium  would  not  take 
place,  because  the  force  AD  could  not  be  decomposed  into  forces 
passing  through  C  and  C ' ,  without  a  third  arising  which  would  not 
be  counterbalanced. 

198.  Hence,  if  a  body  which  touches  a  plane  in  one  or  in 
several  points,  be  urged  by  several  forces  directed  at  pleasure,  it  is 
necessary,  (1.)  That  these  forces  should  admit  of  being  reduced  to  a 
single  one  perpendicular  to  the  plane ;  (2.)  That  this,  in  the  case 
where  it  does  not  pass  through  one  of  the  points  of  contact,  should 
be  capable  of  being  decomposed  into  as  many  forces  parallel  to  it, 
as  there  are  points  of  contact,  and  that  these  should  pass  each 
through  one  of  the  points  of  contact. 

199.  If  the  single  force  which  urges  a  body  be  gravity,  it  is 
necessary  that  the  plane  should  be  horizontal  ;  and  if  the  vertical 
plane,  drawn  through  the  centre  of  gravity  of  the  body,  do  not 
pass  through  one  of  the  points  of  contact,  it  is  necessary,  at  least, 
that  it  should  not  leave  all  the  touching  points  on  the  same 
side. 


118  Statics. 

200.  If  therefore,  the  body  be  urged  only  by  two  forces,  it 
is  necessary;  (1.)  That  the  two  forces  should  be  in  the  same 
plane  ;  (2.)  That  this  plane  should  be  perpendicular  to  that  on 
which  the  body  rests ;  (3.)  That  the  resultant  (which  must  be 
always  perpendicular  to  this  last  plane)  should  not  leave  all  the 
points  of  contact  on  the  same  side ;  and  if  one  of  these  forces  be 
gravity,  it  is  necessary,  moreover,  that  this  plane  should  be  vertical 
and  pass  through  the  centre  of  gravity  of  the  body. 

201.  Let  us  now  see  what  ratio  must  exist  between  two  forces 
which  hold   a  body  in  equilibrium  upon  a  plane,     het  F  q,  F  p, 

Fig.lii.be  the  directions  of  these  two  forces,  and  .^S  the  intersection  of 
the  plane  of  these  forces  with  that  upon  which  the  body  rests  ; 
having  drawn  the  perpendicular  FH  upon  AB,  let  us  suppose 
that  on  this  line,  as  a  diagonal,  and  upon  F  q,  F p,  as  sides,  the 
parallelogram  FEDC  is  constructed.  In  order  that  the  resultant 
of  the  two  forces  q  and  p  may  be  directed  according  to  FD  or  FH, 
it  is  necessary  tbat  the  two  forces  q  and  p  should  be  to  each  other 
as  FC  to  FE;  and  then  the  two  forces  j^  and  q,  and  the  pressure 
which  they  exert  upon  the  plane,  and  which  I  shall  represent  by  §, 
will  be  such  as  to  give  the  proportion 

33  q:p:^::FC:FE:  FD. 

202.  According  to  article  48,  we  have  likewise 

q  '.  p  :  Q  ::  sm  EFD  :  sin  CFD  :  sin  EFC. 

203.  From  the  two  points  ,/l,  B,  taken  arbitrarily  in  AB,  we 
let  fall  upon  the  directions  of  the  two  forces  q,  p,  the  perpendicu- 
lars AG,  BG.  The  triangle  ABG  having  its  sides  perpendicular 
respectively  to  those  of  the  triangle  FDE,  the  two  triangles  will  be 
similar ;  hence 

AG  :  BG  :  AB  ::  DE    or     FC  :  FE  :  FD  :  i  q  :  p  :  g; 

accordingly 

AG  :  BG  :  AB  :  :  q  :  p  :  g. 
But 
Trig.32.       AG  :  BG  :  AB  ::  sin  ABG  :  sin  BAG  :  sin  AGB, 

therefore 


Inclined  Plane.  119 

q  :  p  :  ^  ::  sm  ABG  :  sin  BAG  :  sin  AGS-, 

that  is,  when  two  forces  only  act  upon  a  body  to  retain  it  in  equi- 
librium upon  a  plane  ;  if  we  imagine  two  other  planes  to  which  the 
forces  are  perpendicular,  these  two  forces  and  the  pressure  upon 
the  given  plane,  are  represented  each  by  the  sine  of  the  angle  com- 
prehended between  the  planes  to  which  the  two  other  forces  are 
perpendicular. 

204.  Since  the  ratios  which  we  have  established,  take  place 
whatever  be  the  nature  of  the  two  forces  p   and  q,  they  will  hold  Fig.il2. 
true  when  one  of  the  forces,  p  for  example,  is  gravity  ;  in  this  case 

the  plane  BG  \s  horizontal,  and  the  intersection  BG  is  called  the 
base,  and  AL,  perpendicular  to  BG,  the  height  of  the  plane. 

205.  Since  by  article  202, 

q  :  p  :  ^  ::  sin  EFD  :  sin  CFD  :  sin  EFC, 

we  have 

q  :  p  ::  sm  EFD  :  sin  CFD, 

:  :  sin  HF  p  :  sin  HF  q  ; 

if,  therefore,  knowing  the  weight  p,  the  power  q,  and  the  angle 
HF p,  which  the  direction  of  the  weight  p  makes  with  the  per- 
pendicular to  the  plane,  we  would  determine  the  angle  which 
the  direction  of  the  power  q  must  make  with  the  same  per- 
pendicular, we  shall  obtain  it  by  the  above  proportion,  which 
gives 

sin  HFq=P>^''''^^P. 

But  when  an  angle  is  determined  by  its  sine,  there  is  no  reason 
for  taking  as  the  value  of  this  angle,  the  angle  itself  found  in  the 
tables,  rather  than  its  supplement.  Accordingly,  the  same  weight  Trig.  13. 
may  be  supported  upon  the  same  plane,  by  the  same  power,  di- 
rected in  two  different  ways.  These  two  directions  must  there- 
fore be  such  that  the  two  angles  HF  q,  HF  q,  which  they  form 
with  the  perpendicular  FH,  may  be  supplements  to  each  other. 
Now  if  we  produce  the  perpendicular  HF,  toward  /,  the  greater 
of  these  two  angles  HF  q  is  the  supplement  of  q  FI;  therefore, 
since  it  must  also  be  the  supplement  of  the  smaller  angle  HF  q, 
it  follows  that  q  FI  is  equal  to  the  smaller  angle  HF  q.     Hence 


120  Statics. 

the  two  directions  according  to  which  the  same  power  will  sus- 
tain a  given  weight  upon  the  same  plane,  are  equally  inclined  with 
respect  to  a  perpendicular  to  this  plane,  and  consequently  with 
respect  to  this  plane  itself  j  and  they  both  fall  on  the  side  of  a 
perpendicular  to  this  plane,  opposite  to  that  in  which  the  gravity  of 
the  body  is  directed. 

206.  In  the  same  proportion, 

q  :  p  :  :  sin  HF p  :  sin  HF  q, 

if,  instead  of  the  angle  HF  p,  we  put  the  inclination  ABG  of  the 
Geom.    plane,  which  is  equal  to  this  angle,   and  instead  of  sin  HF  q,  its 
equal  cos  A'F  q^    FA'   being  drawn  parallel   to   BA,   we   shall 
have 

q  :  p  :  :  s\n  ABG  :  cos  A'F  q, 


209 


and  hence 


p  X  sin  ABG 

^  cos  A'F  q 


Therefore,  the  inclination  of  the  plane  and  the  weight  remaining 
the  same,  the  power  q  must  be  so  much  the  smaller,  as  the  cosine 
of  its  inclination  to  the  plane  is  greater;  accordingly,  as  the 
Fig.  113.  greatest  of  all  the  cosines  is  that  of  0°,  we  say  that  the  direction  in 
which  a  power  acts  to  the  greatest  advantage,  in  sustaining  a  weight 
upon  an  inclined  plane,  is  that  which  is  parallel  to  this  plane. 

207.  In  this  case  the  proportion 

q  :  p  :  :  sm  ABG  :  cos  A'F  q 
becomes 

q  :  p  :  :  sin  ABG  :  1  or  radius. 

Fig.113.  Now  if,  from  the  point  A,  we  let  fall  the  perpendicular  AL  upon 
the  horizontal  line  BG,  we  shall  have  in  the  right-angled  triangle 
ALB, 

Trig.  30.  sm  ABG  :  1   :  :  AL  :  AB  ; 

therefore 

q  :  p  ::  AL  :  AB; 

that  is,  ivhen  the  power  acts  in  a  direction  parallel  to  the  plane,  ii 
is  to  the  weight  as  the  height  of  the  plane  is  to  its  length. 


Inclined  Plane.  121 

208.  If  the  direction  of   tRe  power  be  horizontal,   the  angle  Fig.ii4. 
A'F  q,  being  the  complement  of  BAL,  the  proportion  becomes 

q  :  p  :  :  sin  ABG  :  cos  A'F  q, 
q  :  p  :  :  sin  ABG  :  sin  BAL, 

::      AL         :     BL;  Trig.  32. 

that  is,  when  the  direction  of  the  poiver  is  parallel  to  the  base  of  the 
inclined  plane,  the  power  is  to  the  weight  as  the  height  of  the  plane 
to  its  base. 

From  the  proportion 

q  :  p  :  :  sin  ABG  :  cos  A'F  q, 

we  infer,  as  a  general  conclusion,  that  so  much  less  power  is  re- 
quired according  as  the  inclination  of  tiie  plane  is  less,  and 
according  also  as  the  inclination  of  the  power  to  the  plane  is 
less. 

We  have  said  nothing  of  the  point  where  the  direction  of  the 
power  is  to  be  applied  to  the  body.  This  point  is  determined  only 
by  the  condition,  that  the  direction  of  the  power  meet  the  vertical 
drawn  througli  the  centre  of  gravity  of  the  body  in  a  point  from 
which  a  perpendicular  let  fall  upon  the  plane  has  the  conditions 
mentioned  in  article  19G,  &;c. 

We  hence  see  that  a  homogeneous  sphere  cannot  be  sustained 
upon  an  inclined  plane,  except  when  the  direction  of  the  sustain- 
ing force  passes  through  the  centre  of  the  figure,  which  is  at  the 
same  time  the  centre  of  gravity. 

209.  If  several  powers,  instead  of  one,  are  opposed  to  the 
action  of  the  weight,  what  we  have  said  respecting  the  power  q,  is 
to  be  understood  of  the  resultant  of  these  several  powers.     If  the 

body  p,  for  example,  is  supported  upon  an  inclined  plane  by  the  Fig.ll5. 
combined  action  of  a  power  q,  and  of  the  resistance  of  a  fixed 
point  B,  to  which  is  attached  the  cord  HD  q,  passing  round  the 
body ;  through  the  point  of  meeting  S,  of  the  two  cords  BH,  q  X), 
suppose  a  line  SF  drawn  so  as  to  bisect  the  angle  formed  by  the 
cords.  If  this  line  cut  a  vertical  line  passing  tiirougb  the  centre 
of  gravity  in  a  point  F,  from  which  a  perpendicular  can  be  let 
fall  upon  the  plane  that  shall  pass  through  the  point  of  contact 
Mech.  16 


122  Statics. 

H,  the  equilibrium  will  be  possible  ;  and  the  ratio  of  the  weight 
p  to  the  effort  in  the  direction  SF  will  be  determined  by  the  fore- 
going rules.  The  ratio  of  the  effort,  in  the  direction  SF,  to  the 
power  (7,  will  be  the  same  as  in  the  movable  pulley.  Thus  if 
207.  the  power  q  is  exerted  in  a  direction  parallel  to  the  plane,  the 
weight  p  will  be  to  tiie  power  q,  as  the  length  of  the  plane  to  half 
its  heiglit ;  that  is,  the  power  will  be  only  one  half  of  what  would 
be  necessary  without  the  aid  of  the  pulley,  or  fixed  point  B. 

210.  With  respect  to  the  whole  pressure  exerted  ui.ion  the 
plane,  it  will  be  easily  determined  by  the  ratios  above  establish- 
ed. As  to  the  particular  pressure,  however,  that  takes  place 
upon  each  of  the  points  where  the  body  rests  upon  the  plane,  it 
is  absolutely  indeterminate,  except  in  the  case  where  the  body 
touches  only  in  two  jjoints  ;  and    in  this   case   the   whole   pressiu'e 

161.  is  divided  between  these  two  points  in  the  inverse  ratio  of  the 
distances  of  its  direction  from  tliese  points.  In  every  other  case 
there  are  no  other  conditions  for  determining  the  several  pres- 
sures except  (1.)  That  the  sum  of  them  must  be  equal  to  the 
whole  pressure.  (2.)  That  the  sum  of  their  moments,  taken  with 
respect  to  an  axis  perpendicular  to  the  direction  of  the  whole 
pressure,  is  zero  ;  the  same  will  be  true  of  the  sum  of  the  mo- 
ments with  respect  to  another  axis  perpendicular  to  the  first. 
These  two  axes,  moreover,  pass  through  a  point  in  the  direc- 
tion of  the  whole  pressure.  Thus,  when  a  body  rests  upon  a 
plane  by  means  of  a  plane  surface,  there  is  no  reason  for  suppos- 
ing that  all  the  points  upon  which  it  rests  should  experience 
equal  pressures,  except  when  it  has  the  figure  of  a  right  prism 
or  a  right  cylinder. 

211.  With  respect  to  bodies  which  rest  upon  several  planes  at 
once,  either  in  virtue  of  a  single  force,  or  of  several  forces,  in 
which  we  comprehend  their  gravity,  the  general  law  of  equilibrium 
is,  (1.)  That  the  resultant  of  all  these  forces  must  admit  of  being 
decomposed  into  as  many  forces  as  there  are  points  on  which  the 
body  rests  ;  (2.)  That  these  must  be  perpendicular  to  the  plane 
touching  the  body  at  this  point. 

Let  a   heavy   body  KGI  be   placed   in  equilibrium  upon  two 

Fig.  116. inclined  planes;  this  state  can  continue  only  while   the  weight  of 

the  body  is  destroyed  by  the  resistance  of  the  planes ;  if  there- 


Inclined  Plane.  123 

fore  the  body  is  in  contact  with  each  of  the  planes  only  in  a  sin-* 
gle  point,  and  perpendiculars  10,  KO,  be  drawn  through  these 
points,  they  must  meet  in  some  common  point  O,  of  the  vertical 
passing  through  the  centre  of  gravity  G,  in  order  that  the  weight 
of  the  body  may  admit  of  being  decomposed  into  two  other  forces 
having  directions  perpendicular  to  these  j)lanes.  The  compo- 
nents 10,  KO,  will  represent  the  pressures  exerted  upon  the 
planes.  It  hence  results,  that  the  plane  ivhich  passes  through  the 
points  of  support  and  the  centre  of  gravity  must  he  vertical,  or  per- 
pendicular to  the  inclined  planes,  or  to  their  common  intersection, 
which  will  consequently  be  horizontal. 

What  is  here  said  is  not  peculiar  to  the  case  of  a  body  urged 

by  gravity  simply.     Whatever  be  the  forces  acting  at  /,  K,   their 

resultant    must   conform    to    what   we   have    said    of  the    vertical 
passing  through  the  centre  of  gravity. 

Let  XZ  be  a  horizontal  plane  passing  through  the  intersec- 
tion B  of  the  inclined  planes  ;  and  through  the  point  K,  draw  KH 
also  horizontal ;  and  let  the  weight  of  the  body  KIG  be  repre- 
sented by  Q,  and  the  pressures  exerted  upon  the  two  planes  AB, 
BC,  by  p,  q,  respectively.  In  order  to  obtain  these  pressures, 
we  must  suppose  the  weight  g  of  the  body  to  be  a  vertical  force 
applied  at  O ;  thus  regarded,  it  may  be  decomposed  into  two 
others,  directed  according  to  01,  OK;  we  have  accordingly  the 
following  proportions, 

9  :  p  :  (^  :  :  sin  lOK  :  sin  GOK  :  sin  lOG,  48. 

or,  since  the  angle  CBZ  =  GOK,  and  ABX  =  lOG,  and  the 
angles  IBK,  lOK,  are  supplements  of  each  other,  Geom. 

80. 
g  :  p  :  q  ::  sxnABC  :  sm  CBZ  :  sin  ABX,  Tng.  13. 

:  :  sin  HBK  :  sin  BKH  :  sin  KHB, 

HK     :        HB     :        BK. 


Trig.  32. 


212.  These  principles  are  sufficient  for  determining,  under 
all  circumstances,  the  conditions  of  equilibrium,  where  planes 
are  concerned.  By  means  of  them  we  are  enabled  to  explain 
the  strength  of  arches,  and  in  general  why  hollow  bodies,  whose 
exterior  surface  is  convex,  are  better  fitted,  on  this  account,  to 
resist  a  compressing  force.     If,  for  example,  a  body  is  composed 


124  Statics. 

Fig.ii6f  of  four  pnrts  ABCD,  CDFE,  FEGH,  ABGH,  the  exterior  and 
interior  surfaces  of  which  are  circular  and  concentric,  and  the 
same  force  be  applied  to  the  centre  of  gravity  of  each  part,  and 
be  directed  toward  the  common  centre  of  the  whole,  no  separa- 
tion can  take  place  among  the  parts,  however  great  the  force 
employed,  provided  the  material  itself  be  sufficiently  hard.  For 
it  will  be  seen,  that  the  force  belonging  to  each  part  may  be  con- 
sidered as  decomposed  into  two  others  perpendicular  respectively 
to  the  two  plane  faces  of  this  part,  and  that  consequently  between 
each  pair  of  contiguous  planes  there  will  be  two  equal  and  directly 
opposite  forces  ;  so  that  the  several  forces  will  mutually  destroy 
each  odier,  and  a  general  equilibrium  will  be  the  result.  The 
parts  ABCD,  &-c.,  are  called  voussoirs.  In  a  regular  arch,  the 
upper  voussoir  is  distinguished  by  the  name  of  key-stone.  The 
surfaces  which  separate  the  voussoirs  are  techtiically  termed 
joints.  The  interior  curve  of  the  arch  is  called  the  inirados,  and 
the  exterior,  or  that  which  limits  all  the  voussoirs,  when  they  are 
in  equilibrium,  is  called  the  extrados ;  the  masses  of  masonry  at 
each  end,  that  support  the  arch,  are  the  abutments.  The  begin- 
ning of  the  arch  is  called  the  spring,  the  middle  the  crown,  and  the 
parts  between  the  spring  and  the  crown,  the  haunches  of  the  arch. 
The  part  of  the  abutment  from  which  the  arch  springs,  is  termed 
the  impost ;  and  the  distance  between  the  imposts  the  span  of  the 
arch. 

Of  the  Screw. 

Fiffin         ^^o.  The  screw  AB,  is  a  solid   cylinder  having  a  protuberance 
118.       or    thread    raised    upon    its    convex    surface,    and    carried    round 
obliquely,  and  continually  with  the  same  inclination  to  the  axis. 

The  nut  is  a  hollow  cylinder  with  a  spiral  groove  cut  upon 
the  concave  surface,  and  fitted  to  receive  the  thread  of  the  screw. 
The  former  is  sometimes  called  the  external,  and  the  latter  the 
internal  screw. 

Sometimes  the  nut  is  fixed,  and  the  screw  iu  turning  has  all  its 
threads  carried  successively  through  it ;  sometimes  the  screw  is 
fixed,  and  the  nut  in  turning  passes  the  whole  length  of  the  screw. 
In  each  case,  while  the  power  is  applied  at  the  same  distance  from 
the  axis  of  the  screw,  there  is  always  the  same  ratio  between  this 


Scretv.  125 

power  and  the  force  which  it  is  capable  of  exerting  in  the  direction 
of  the  axis. 

214.  We  shall   have  a  pretty  just  idea  of  the  screw,  by  rep- 
resenting  the   thread   as  formed   by   wrapping    round   the   cylinder 

the  hypothenuses  CK of  as  many  right-angled  triangles  CIK,  asFig.119. 
there  are  revolutions  of  the  thread,  each  triangle  having  for  its 
height,  the  distance  CI  between  two  adjacent  threads,  and  for  its 
base  IK,  the  circumference  of  the  cylinder  corresponding  to  the 
point  /;  so  that,  according  as  the  thread  becomes  thicker,  IK  is 
increased  in  length,  the  height  C/ remaining  the  same. 

In  figure   118,  where  the  threads  are  edge-shaped,   according 
as  the   protuberant   part  becomes  thicker,  or  departs  farther  from 
the  axis,  we  must  suppose  that  the  base   IK  increases,  and  that  the  Fig.  119. 
height  C/ diminishes. 

215.  The  screw  AB  being  fixed,  and  having   a  vertical   posi- 
tion, no  allowance   being  made   for  fiiction,  or  for  the   nut    hav- 
ing its  natural  gravity,   it  is  evident  that  the  nut  in  turning  would  Fig.  117, 
pass  over  the  several  threads  of  the  screw  by   sliding   upon   each  ^^^' 

as  upon  an  inclined  surface.  It  is  also  evident  that  this  tenden- 
cy may  be  overcome  by  applying  to  the  nut  XZ,  a  certain  pow- 
er, which  admits  of  being  directed  in  several  different  ways. 
But  as  the  nut  has  manifestly  no  motion,  if  it  be  prevented  from 
turning,  we  shall  confine  ourselves  to  inquiring  what  must  be  the 
ratio  between  the  weight  of  the  nut,  or  in  general  between  the 
force  which  urges  it  in  a  direction  parallel  to  the  axis  of  the  screw, 
and  the  force  capable  of  preventing  its  turning.  Of  the  several 
points  of  the  nut,  wc  shall  first  consider  only  that  which  rests  upon 
one  point  of  the  thread  of  the  screw. 

The  force  which  acts  immediately  on  this  point  to  prevent 
the  turning,  and  that  which  tends  to  make  it  descend  parallel  to 
the  axis,  must  be  regarded  as  being  in  equilibrium  upon  an  in- 
clined plane  whose  height  is  the  perpendicular  distance  between 
two  adjacent  threads,  and  whose  base  is  the  circumference  of  the 
circle  which  would  be  described  by  the  point  in  question.  This  fol- 
lows from  what  we  have  said  of  tbc  nature  of  the  screw.  Now  of 
these  two  forces,  the  first  is  parallel  to  the  base  of  the  inclined  plane, 
and  the  second  is  perpendicular  to  it ;  hence,  by  article  208,  it  will 


126  Statics. 

be  seen  that  the  part  of  the  force  parallel  to  the  axis  of  the 
screw  which  is  exerted  upon  any  point  of  the  thread,  is  to  the 
force  which  it  is  necessary  lo  apply  at  this  point  lo  prevent  die 
turning,  as  the  base  of  the  inclined  plane  is  to  its  height,  that  is, 
as  the  circumference  of  the  circle,  which  would  be  described  by 
the  point  of  application,  is  to  the  perpendicular  distance  between 
two  adjacent  threads.  Therefore,  if  we  call  p  the  first  force,  and 
q'  the  second,  and  d  the  distance  of  the  point  of  application  from 
the  axis,  h  the  height  of  the  supposed  inclined  plane  or  perpen- 
dicular distance  between  two  adjacent  threads,  and  2  n  the  ratio  of 
the  radius  of  a  circle  to  its  circumference,  we  shall  have 
p  :  q'  :  :  2  d  7t  :  h. 

But  each  point  of  the  nut  is  not  supported  directly ;  the 
whole  is  subjected  to  a  certain  power  q,  applied  at  some  point  of 
the  nut  whose  distance  from  the  axis  may  be  represented  by  D. 
It  is  hence  evident,  that  D  being  greater  than  d,  there  will  be 
necessary  for  each  point,  a  force  so  much  the  less,  according 
as  the  distance  D  is  greater ;  so  that  if  we  call  q  the  part  of  this 
force  which  at  the  distance  D  is  capable  of  the  same  effort  as  q'  is 
at  the  distance  5,  we  shall  have 

q'  :  q  :  :  D  :  d. 

MuUiplying  this  proportion  by  the  former,  we  shall  have 

p  :  q  :  :  2  71  D  d  :  h  d  :  :  2  n  D  :A; 

that  is,  for  each  point  of  the  nut  that  rests  upon  the  thread  of  the 
screw,  there  is  the  same  ratio  between  the  force  exerted  parallel 
to  the  axis,  and  that  which  at  a  given  distance  D  prevents  the 
turning  ;  and  this  ratio  is  that  of  2  ti  jD  to  A.  Now  2  tt  D  is  the 
circumference  of  the  circle  which  would  be  described  by  the 
power  q  in  turning ;  we  conclude,  therefore,  that  the  sum  of  all 
the  forces  p  which  urge  the  nut  parallel  to  the  axis,  is  to  the  sum 
of  all  the  powers  5-  necessary  to  prevent  the  turning,  as  the  cir- 
cumference of  the  circle  which  would  be  described  by  the  pow- 
er q,  is  to  the  distance  between  two  adjacent  threads  of  the 
screw. 

216.  Hence  the  force  which  it  is  necessary  to  employ  par- 
allel to  the  axis  of  the  screw,  to  prevent  the  power  q  from  turning 
the  nut,  must  be  to  this  power  q,  as  the  circumference  which  this 


Screw.  1 27 

power  tends  to  describe,  is  to  the  distance  between  two  adjacent 
threads. 

217.  Therefore,  upon  the  same  screw,  the  effect  of  the  power 
q  will  be  so  much  the  more  considerable  according  as  it  is  ap- 
plied at  a  greater  distance  from  the  axis ;  and  upon  different 
screws,  the  power  being  applied  at  the  same  distance  from  the 
axis,  the  effect  will  be  so  much  the  more  considerable  according 
as  the  distance  between  the  threads  is  less. 

218.  The  screw,  therefore,  is  a  compound  machine  partaking  of 
the  inclined  plane  and  the  lever ;  it  is  advantageously  employed  in 
compressing  bodies  and  for  several  other  purposes.  Friction, 
however,  greatly  modifies  tlie  effect  which  this  machine  ought  to 
produce  according  to  the  ratio  above  established. 

219.  In  order  that  the  nut  may  pass  through  the  distance 
between  two  adjacent  threads,  it  is  necessary,  as  we  have  seen, 
that  the  power  should  make  an  entire  revolution.  This  condi- 
tion is  unalterable,  and  there  are  many  occasions  on  which  an 
important  use  may  be  made  of  it.  When  it  is  proposed,  for  ex- 
ample, to  measure  the  different  parts  of  a  very  small  space  -45,  Fig.120.- 
it  may  be  done  by  causing  this  space  to  be  described  by  the 
extremity  E  of  a  screw  DE,  the  threads  of  which  are  accurately 
formed  at  the  same  distance  from  each  other  throughout.  If  this 
screw  be  made  to  carry,  at  its  other  extremity,  a  dial  GIH,  the  di- 
visions of  which,  as  the  screw  turns,  pass  under  the  fixed  index  FI; 
having  ascertained  what  number  of  turns  the  screw  must  make  in 
order  that  the  point  E  shall  describe  the  known  extent  AB,  we  shall 
be  able,  by  the  number  of  revolutions  and  parts  of  a  revolution 
performed  in  causing  the  point  E  to  pass  over  any  part  of  AB^ 
to  determine  the  length  of  this  part,  hovi'-ever  small  it  may  be* 
If  for  instance  the  distance  of  the  threads  asunder  be  one  tenth 
of  an  inch,  and  the  >pircumference  of  the  dial  GIH  five  inches, 
any  point  of  the  circumference  will  move  through  five  inches 
while  the  point  E  advances  one  tenth  of  an  inch  ;  consequently, 
the  circumference  being  divided  into  tenths  of  an  inch,  while  one 
division  passes  under  the  index,  the  point  E  would  be  carried 
forward  only  -^^  of  yV>  ^^  sis  ^^  ^^  "^ch.  This  is  called  a  »u- 
crometer  screw. 


128  Statics. 

220.  By  applying  tlie  screw  to  other  machines  their  effect  is 
greatly  increased.  If  the  power  q,  for  example,  applied  to  the 
Fig.121.  winch  DE  q,  is  made  to  turn  the  screw  DC,  the  threads  of  which, 
acting  upon  the  teeth  of  the  wheel  fV,  cause  it  to  turn,  and  with 
it  the  cylinder  7,  around  which  passes  the  cord  Kp  carrying  the 
vveiglit  p ;  the  ratio  of  the  power  q  to  the  weight  p,  may  be 
determined  thus.  Calling  q'  the  force  exerted  by  the  thread 
of  tiie  screw  upon  one  of  the  teeth  of  the  wheel  TV,  we  shall 
have 

215.  q  :  q'  :  :  AB  :  circiim.  DE, 

AB  being  the  distance  between  two  threads  of  the  screw,  and 
circum.  DE   denoting  the   circumference  of  the  circle  described 

215.  by  the  power  q.  The  force  q'  is  a  power,  which,  applied  to  the 
circumference  of  the  wheel,  is  exerted   against  the  weight  p  ;  ac- 

186.     cordingly  we  have 

q'  :  p  ::  IK  :  IL, 

and,  by  taking  the  product  of  the  corresponding  terms  of  the  two 
proportions, 

qq'  :  q' p  :  :  AB  X  IK  :  circum.  DE  X  IE, 


or. 


p     :  :  AB  X  IK  :  circum.  DE  X  IE ; 


by  which  it  will  be  seen  that  q  has  so  much  the  more  advantage 
according  as  AB  and  IK  nve  smaller,  considered  with  reference  to 
circum.  DE  and  IE.  DC  in  this  case  is  called  a  perpetual 
screw. 


Of  the   Wedge, 

Fig.122.  221.  The  wedge  is  a  triangular  prism  intended  to  be  intro- 
duced into  a  cleft  for  the  purpose  of  enlarging  it,  or  between  two 
surfaces,  in  order  to  separate  them  further  from  each  other,  or  to 
fix  them  at  a  determinate  distance. 

The    action    of   the   wedge,   considered   as  an  instrument   for 
cleaving,  is  essentially  modified  by  friction  and  other  causes.     As 


Wedge.  129 

there  are  no  bodies  which  have  not  a  certain  degree  of  flexibihly, 
the  parts  of  the  cleft  in  contact  with  the  faces  of  the  wedge  may- 
be separated  further  from  each  other  without  the  extremity  Z  of 
the  cleft  .shifting  its  place  ;  so  that  a  part  of  the  force  applied  at  the 
back  DE  of  the  wedge  is  employed  in  simply  bending  the  two 
branches  which  form  the  cleft  j  and  the  other  is  exerted  in  distend- 
ing the  fibres  of  the  part  that  has  not  yet  yielded. 

222.  As  this  resistance  depends  upon  causes  so  numerous  and 
so  variable  at  the  same  time,  it  is  not  to  be  expected  that  the  na- 
ture and  operation  of  the  wedge,  considered  physically,  will  ever 
be  reduced  to  a  clear  and  satisfactory  theory.  In  a  mathematical 
point  of  view,  the  following  explanation  seems  to  be  unexceptiona- 
ble. 

223.  We  suppose  the  direction  of  the  power  p,  to  be  perpen- F'g- 123. 
dicular  to  the  back  of  the  wedge,  since  if  it  is  not,  it  may  always 

be  decomposed  into  two  others,  one  perpendicular,  and    the  other 
parallel  to  the  back,  of  which  the  latter  is  incapable  of  urging  the    45. 
wedge  backward  or  forward.     This  perpendicular  force,  therefore,    35. 
may  be  considered   as  keeping  the   wedge  JIBC  in  equilibrium, 
while  pressed  at  /,  K^   by  the  parts  of  a  body  that  tend  to  unite. 
The   theory   of   the   inclined    plane    is    accordingly    applicable  to 
this   case,    and   the   resistance    exerted    at  J,  K,    cannot    destroy     211. 
the    action    of  the   power  />,   except   while  this   power    admits  of 
being    decomposed    into    two    others  q,  r,   passing    through  these 
points,  and  directed  perpendicularly  to  the  faces  BC,  AC,  of  the 
wedge.     Therefore,  the  forces  p,  q,  r,  must  meet  in  the  same 
point  E,  be  in  the  same  plane  ABC,  and  have  the  following  pro- 
portion to  each  other,  namely, 

p  :  q  :  r  :  :  sin  q  E  r  :  s\n  p  E  r  :  s\n  p  E  q, 

or,  since  the  sines  of  the  angles   q  E  r,  p  E  r,  p  E  q,  are  equal  Geom. 
respectively  to  the  sines  of  their  supplements  C,  A,  B,  ^^.    jg 

p  :  q  :  r  :  :  sm  C  :  sin  A  :  sm  B, 
:  :      AB  :     BC     :      AC, 

that  is,  the  three  forces  p,  q,  r,  are  to  each  other  as  the  three  sides 
of  the  triangle  to  which  their  directions  are  perpendicular. 
Mech.  17 


130  Statics. 

224.  The  three  straight  lines  AB,  BC,  AC,  are  to  each  other 
as  the  faces  of  the  wedge  to  which  they  respectively  belong,  for 
these  faces  are  parallelograms  of  the  same  base,  and  whose  alti- 

Geom.  tildes  are  AB,  BC,  AC ;  it  follows,  therefore,  that  the  power  p 
and  its  two  components,  are  to  each  other  as  the  back  and  two 
sides  of  the  wedge,  or  in  other  words,  that  the  power  being  repre- 
sented by  the  back  of  the  wedge,  the  force  exerted  against  the  two 
sides  ivill  be  represented  by  these  sides  respectively.  A  very  acute 
wedge,  therefore,  or  one  whose  sides  are  very  long  compared  with 
the  back,  possesses  an  advantage  in  the  same  proportion,  and  may 
be  made  to  exert  a  great  power  by  means  of  a  very  moderate  blow 
on  the  back. 

225.  There  has  not  been  a  perfect  agreement  among  mechan- 
ical writers  as  to  the  theory  of  the  wedge.  The  direction  of  the 
resistance  has  sometimes  not  been  sufficiently  attended  to,  and 
the  circumstance  of  one  of  the  resistances  proceeding  from  an 
immovable  obstacle  in  certain  cases,  has  sometimes  been  over- 
looked. 

226.  To  the  wedge  are  referred  all  cutting  instruments,  as 
knives,  scissors,  the  teeth  of  animals,  &^c.  A  saw  is  a  series  of 
wedges  on  which  the  motion  impressed  is  oblique  to  the  resistance. 
A  wimble  is  a  combination  of  the  screw  and  the  wedge. 

To  the  wedge  of  the  pyramidal  form,  are  reduced  all  piercing 
instruments,  as  nails,  bayonets,  stakes,  piles,  he. 


General  Law  of  Equilibrium  in  Machines. 

227.  By  combining  together,  in  different  ways,  the  machines 
above  considered,  we  can  form  others,  the  number  of  which  may 
be  multiplied  without  end.  With  respect  to  compound  machines, 
we  determine  the  ratio  of  the  power  to  the  resistance  necessary  to 
an  equilibrium,  by  having  regard  to  the  tensions  of  the  cords  that 
connect  the  different  parts,  this  ratio  being  supposed  to  be  known 
for  each  of  the  component  parts. 

But  however  complicated  the  machine,  there  is  a  simple  rule 
by  which  the  ratio  of  the  power  to  the  resistance  is  obtained  directly 


Principle  of  Virtual  Velocities.  131 

from  the  ratio  of  the  spaces  which  the  points  of  application  of 
the  two  forces  tend  to  describe  in  the  same  time.  This  is  a  par- 
ticular case  of  what  is  called  the  principle  of  virtual  velocities. 

228.  Let  us  suppose  a  very  small  motion  given  to  the  machine, 
and  that  the  points  of  application  of  the  two  forces,  describe  curves 
to  which  the  directions  of  these  forces  are  tangents.  Let  u  denote 
the  velocity  of  the  force  p,  or  the  space  described  in  any  given 
time  by  p,  and  v  the  corresponding  velocity  of  q,  or  the  space 
described  by  q  in  the  same  time  ;  if  the  ratio  of  q  to  p,  neces- 
sary to  an  equilibrium,  is  required,  we  shall  obtain  it  very  nearly  by 
the  proportion 

q   :  p  :  :  u  :  v; 

and  this  will  approach  so  much  the  more  nearly  to  the  exact  ratio, 
according  as  the  motion  impressed   upon  the  machine  is  less,  so 
that  by  taking  the  limit  of  the  ratio  of  u  to  v,  we  shall  have  exactly  '^"S-  ^^■ 
the  ratio  sought  of  q  to  p. 

If  the  directions  of  the  forces  q,  p,  are  not  tangents  to  the 
curves  described  by  the  points  of  application  of  these  forces,  we 
take,  instead  of  the  spaces  described  by  these  points,  the  projec- 
tions of  these  spaces  upon  the  directions  of  the  forces,  and  the 
inverse  ratio  of  these  projections  will  be  the  ratio  of  the  forces, 
in  the  case  of  an  equilibrium.  We  are  at  liberty  to  take  for  the 
point  of  application  of  each  force,  any  point  we  may  choose  in 
its  direction,  provided  we  regard  it  as  firmly  attached  to  the 
machine. 

229.  We  shall  now  apply  this  rule  to  a  few  examples  in  order 
to  show  its  truth  and  utility. 

In  the  lever,  I  take  for  the  points  of  application  of  the  forces 
p,  q,  the  feet  L,  M,  of  the  perpendiculars  FL,  FM,  let  fall  from  Fig.  124. 
the  fulcrum  jF,  upon  the  directions  of  the  forces.  When  the  lever 
turns  about  the  point  F,  the  points  L,  M,  will  describe  the  similar 
arcs  LD,  AIM',  to  which  the  directions  p  L,  q  M,  of  the  forces, 
are  tangents.  The  lengths  of  these  arcs  are  to  each  other  as  their 
radii  FL,  FM  j  so  that  we  have  the  proportion,  Geom. 

LD  :  MM  ::  FL  :  FM,  ^ 

that  is, 

u      :      V       ::  FL  :  FM; 


132  Statics. 

and  as  this  ratio  remains  the  same,  however  small  the  motion  im- 
pressed upon  the  lever,  it  holds  true,  when  u  :  v  :  :  q  :  p,  which 
gives  exactly 

q   :  p   ::  FL  :  FM. 

In  the  case  of  an  equilibrium,  therefore,  the  two  forces  are  to  each 
other  in  the  inverse  ratio  of  the  perpendiculars  let  fall  upon  their 
directions,  as  already  determined  by  a  different  method. 

230.  If  we  had  taken  the  extremities  B,  D,  of  the  lever  for 
the  points  of  application  of  the  forces  p,  q,  the  directions  of  the 
forces  would  no  longer  be  tangents  to  the  arcs  described  by  these 
points.  We  should  therefore  have  to  project  these  arcs  upon  the 
straight  lines  Bp,  D  q,  then  to  take  the  ratio  of  these  projections, 
and  seek  the  limit  of  this  ratio. 

On  the  supposition  of  motion,  the  angles  DFD',  BFB',  de- 
scribed by  the  arms  FB,  FD,  are  equal ;  and  the  arcs  BB',  DD', 
described  by  B,  D,  about  the  point  F,  as  a  centre,  are  to  each 
other  as  the  radii  FB,  FD.  This  ratio  continues  the  same  when 
the  arcs  become  infinitely  small,  so  that  we  have  constantly 

FB  :  BE'  :  :  FD  :  DD'. 

From  the  points  B',  D',  let  fall  the  perpendiculars  B'Jl,  D'C,  upon 

the  directions  of  the  forces  p,  q  ',  and  we  shall  have 

u  =  BA,         and         v  =  DC. 

Also  from  F,  let  fall  the  perpendiculars  FM,  FL,  upon  the  direc- 
tions of  the  forces.  By  *  considering  the  infinitely  small  arcs 
Geom.  BB',  DD",  as  straight  lines,  perpendicular  respectively  to  the  radii 
FB,  FD,  the  triangles  DD'C,  FMD,  are  similar,  as  also  the  tri- 
angles BB'A,  FLB ;  whence 


209 


FB  :  FL  ::  BB'  :  BA  =  -^  X  FL, 


and 


FD  :  FM  ::  DD'  :DC  =  ^^  X  FM ; 

accordingly  we  have,  by  substitution, 

BB'         „r        ,  DD'  „,,, 

u  =  -j^  X  FL,  and  v  =  -^^   X  FM, 


Principle  of  Virtual  Velocities.  133 

X  FM, 


and  hence 

u  : 

V  :  :  ■ 

BB' 
FB 

-XFL  : 

DD' 
FD    ^ 

that 

is,  since 

BB' 
FB 

=  - 

DD' 
FD  ' 

u  : 

V  :  : 

FL 

:  FM', 

from 

I  which  we 

obtain 

1  as  before, 

9  '• 

p  :: 

FL 

:  FM. 

Geom. 
288. 


231.  In  the  wheel   and   axle,   when  motion   commences,   theFig.i02. 
points  of  application  of  the  power  and  resistance   describe  similar 

arcs,  or  arcs  of  the  same  number  of  degrees,  the  one  upon  the 
circumference  of  the  wheel,  and  the  other  upon  that  of  the  axle. 
The  directions  of  the  forces  are  tangents  to  these  arcs,  whose 
lengths  are  to  each  other  as  the  radii  IB,  iJl,  of  the  axle  and 
wheel ;  in  this  machine,  therefore,  we  have 

u  :  V    ::  IB  :  M, 

and  accordingly 

q  :  p  :  :  IB  :  M. 

232.  In  the  pulley,  if,  while  the  weight  p  describes  in  rising 

a  space  equal  to  u,  each  of  the  two  cords  which  meet  at  the  mova-  pig.  95. 
ble  pulley,  is  shortened  by  the  same  quantity  u,  the  cord  to  which 
is  suspended  the  power  q,  will  be  lengthened  by  a  quantity  equal 
to  2  u,  which  will  consequently  be  the  space  passed  through  by  q 
in  its  descent.  Taking,  therefore,  v  ==-2  u,  we  shall  have,  in  the 
case  of  an  equilibrium, 

q  :  p  :  :  u  :  V, 
::   1  :  2; 

or  generally,  the  cords  being  parallel, 

q  :  p  ::  1   :  n, 

n  denoting  the  number  of  cords  that  meet  at  the  movable 
pulley. 

I  will  take,  as  the  last  example,  the  assemblage  of  pulleys, 
represented  in  figure  87.  If  the  power  q,  in  descending,  pass 
over    a   space   t^,  the    point    JV  will  be    elevated    by    a   quantity 


134  Staiks. 

equal  to  |  u ;  calling  this  v',  the  point  M  will  be  elevated  by  a 
quantity  equal  to  ^  v'  or  ]  v ;  this  being  designated  by  v",  the 
point  L  will  be  elevated  by  a  quantity  |  v"  or  |  u  ;  and  this  is  the 
space  through  which  the  weight  p  passes  in  rising ;  calling  this  n, 
therefore,  we  shall  have,  whatever  v  may  be,  m  =  i  u,  which 
gives  in  the  case  of  an  equilibrium 

q  :  p  :  :  u  :  V, 
:  :  1   :  8  ; 

or  generally 

9  :  p  :  :  1   :  2", 

n  denoting  the  number  of  movable  pulleys  in  the  system,  which 
agrees  with  what  has  been  before  shown. 

233.  In  the  screw,  when  the  nut  passes  over  a  space  equal  to 
the  distance  between  two  adjacent  threads,  or  when  it  is  elevated 
Fig.118.  through  a  height  equal  to  DE,  the  point  to  which  the  power  is 
applied  describes  a  spiral  about  the  axis  AB,  and  rises  through  a 
space  equal  to  DE,  the  projection  of  which  spiral  upon  a  horizontal 
plane  is  a  circle,  of  which  BG  is  the  radius.  Moreover,  these 
motions  of  the  nut,  of  the  point  of  application,  and  its  projection  are 
such  that  if  the  nut  describes  one  half,  one  third,  or  any  other 
part  of  the  distance  between  two  threads,  the  point  of  application 
will  describe  a  similar  part  of  the  length  of  the  spiral,  and  its  projec- 
tion a  similar  part  of  a  circumference  of  which  BG  \s  the  radius; 
accordingly,  if  we  call  u  the  height  through  which  the  nut  rises,  and 
V  the  arc  of  a  circle  described  in  the  same  time  by  the  horizontal 
projection  in  question,  we  shall  have 

u  :  V  :  :  DE  :  circum.  BG. 

Now  the  direction  of  the  force  q  not  being  a  tangent  to  the 
spiral,  it  is  necessary,  in  order  to  apply  to  this  case  the  general  law 
of  equilibrium,  to  consider  the  projection  of  a  very  small  arc  of  the 
spiral  upon  the  direction  of  the  force  q ;  but  this  direction  being 
supposed  to  be  a  tangent  to  the  circumference  above  mentioned,  the 
projection  upon  this  tangent  will  be  very  nearly  equal  to  the  projec- 
tion upon  the  circumference,  and  one  may  be  taken  for  the  other, 
when  we  consider  only  an  infinitely  small  motion  of  the  nut ;  conse- 
quently, the  ratio  of  the  force  q,  to  that  of  p,  will  be  given,  by 


Friction,  135 

taking  the  limit  of  the  ratio  of  u  to  v.  Now  as  this  ratio  is  constant- 
ly the  same,  and  equal  to  the  ratio  oi  DE  to  circum.  BG  ;  accord- 
ing to  the  preceding  proportion,  we  shall  have,  in  the  case  of  an 
equilibrium, 

q  :  p  :  :  DE  :  circum.  BG, 

as  before  shown  by  a  different  process. 


Of  Friction. 

234.  The  surfaces  of  bodies,  even  the  most  smooth,  are 
covered  with  elevations  and  depressions ;  and  when  two  bodies  are 
brought  in  contact  with  each  other,  the  prominent  parts  of  the  one 
enter  the  cavities  of  the  other,  and  they  cannot  be  moved  the  one 
over  the  other,  without  employing  a  certain  force.  The  resistance 
arising  from  this  cause  is  cd\\e6.  friction. 

There  are  two  sorts  of  friction,  one  which  takes  place  when  the 
bodies  in  contact  have  simply  a  sliding  motion  ;  the  other  when  one 
or  both  the  bodies  move  by  turning  upon  an  axis.  We  have  an 
instance  of  the  former  in  the  motion  of  skates  and  sledges,  and  of 
the  latter  in  the  action  that  exists  between  the  wheels  of  wheel-car- 
riages and  the  ground.  The  resistance  arising  from  the  second 
kind  of  friction,  is  much  less  than  that  of  the  first,  since  a  rotatory 
motion  serves  to  disengage  the  parts  in  contact  without  breaking 
down  the  eminences  or  lifting  them  out  of  the  cavities. 

When  the  rising  up  of  one  body  over  the  other  is  completely 
prevented,  the  friction  becomes  very  intense,  as  appears  in  what 
may  be  accounted  an  extreme  case,  the  drawing  of  wire,  and 
the  rolling  of  bars  of  iron,  copper,  &tc.,  into  plates,  where  the 
extension  of  the  metal  is  the  effect  of  the  friction  acting  all  round  or 
on  two  sides. 

If  the  asperities  with  which  the  surfaces  of  bodies  are  covered, 
were  perfectly  hard,  and  immovably  attached  to  these  surfaces, 
it  would  be  necessary,  in  overcoming  the  friction,  to  raise  the 
incumbent  mass.  If  these  asperities,  on  the  other  hand,  were 
perfectly  flexible,  there  would   be  no  resistance  and  no   friction. 


136  Statics. 

But  as  neither  supposition  is  true  in  any  case,  it  follows,  (1.) 
That  the  resistance  of  friction  arises  in  part  from  the  difficulty 
of  bending  the  asperities,  and  in  part  from  the  necessity  of  rais- 
ing in  a  degree  the  body  or  incumbent  mass ;  (2.)  That,  the 
asperities  having  only  a  limited  degree  of  adhesion,  when  the 
force  necessary  to  cause  the  body  to  move  exceeds  this,  the  asperi- 
ties yield,  or  are  broken  down,  and  the  surfaces  are  gradually 
worn.  Thus  the  effect  of  friction  in  machines  is  not  only  to 
consume  a  part  of  the  force  employed,  but  also  to  destroy  the 
machines  themselves. 

It  would  seem  difficult,  if  not  impossible,  to  establish  general 
rules,  sufficiently  exact  for  determining  the  force  of  friction.  In- 
deed it  will  be  readily  seen,  that  this  resistance  must  vary  according 
to  the  nature  and  texture  of  the  surfaces  in  contact,  their  flexibility, 
and  the  adaptation  in  size  and  figure  of  the  prominent  parts  and 
cavities  to  each  other,  and  according  as  the  force  is  greater  or  less 
by  which  the  surfaces  are  pressed  together  ;  moreover,  on  account 
of  the  flexible  nature  of  surfaces,  the  prominent  parts  are  found  to 
penetrate  to  a  greater  depth  when  more  time  is  allowed  for  enlarg- 
ing the  openings  which  they  tend  to  enter. 

It  belongs  to  experiment  alone  to  enlighten  us  upon  these 
points,  and  to  teach  us  the  proportional  effect  due  to  each.  The 
information,  however,  derived  from  this  source,  is  not  yet  so 
perfect  and  complete  as  could  be  wished,  though  it  is  such  as  may 
be  useful  on  many  occasions.  We  proceed  now  to  make  known 
some  of  the  results  of  experiments,  as  also  the  method  of  applying 
them  in  calculating  the  effect  of  friction  in  the  different  kinds 
of  machines,  and  the  different  kinds  of  motion. 

235.  (1.)  When  the  surfaces  which  are  to  rub  the  one  upon 
the  other,  are  of  the  same  kind  of  matter,  the  resistance  of  fric- 
tion, other  things  being  the  same,  is  greater  than  when  the  sur- 
faces are  of  different  kinds.  Thus,  two  pieces  of  wood  of  differ- 
ent kinds  slide  upon  each  other  with  less  difficulty  than  two  of 
the  same  kind.  Iron  rubbing  on  copper  has  less  friction  than 
iron  on  iron  or  copper  on  copper.  This  is  supposed  to  be  owing 
to  the  prominent  parts  and  cavities  being  more  nearly  fitted  to 
each  other  in  the  latter  case  than  in  the  former. 


Friction.  1 37 

(2.)  The  more  rough  the  surfaces,  or  the  less  they  are  planed 
or  polished,  the  greater  is  the  friction.  This  resistance,  there- 
fore, may  be  diminished,  by  smoothing  the  surfaces,  or  by  fillirg 
the  openings  and  pores  with  other  matter,  as  oil,  soap,  grease, 
black-lead,  he,  with  any  substance  indeed,  which,  while  it  fills 
the  cavities,  does  not  give  rise  to  a  new  adhesion. 

(3.)  It  would  seem  that  the  extent  of  surface  ought  to  contrib- 
ute sensibly  to  the  friction  ;  it  appears,  however,  by  a  great  va- 
riety of  experiments,  that  this  circumstance  makes  but  little 
difference;  we  find  in  fact  the  same  difficulty  for  the  most  part 
in  drawing  a  body  upon  one  of  its  surfaces  as  upon  another, 
though  very  different  in  extent,  provided  they  are  equally  smoothed. 
Thus,  oak  rubbing  on  oak,  is  found  to  have  a  friction  of  about 
44  per  cent. ;  and  on  diminishing  the  surface  as  much  as  possible, 
this  is  reduced  only  to  41^  per  cent.  We  must  except,  however, 
the  case  of  bodies  resting  upon  a  point,  when  the  friction  is 
more  considerable,  than  when  the  contact  takes  place  in  several 
points. 

(4.)  It  is  principally  from  pressure  that  friction  arises,  and 
this  resistance  is  found  to  increase  in  proportion  to  the  pressure ; 
that  is,  we  require  twice  the  force  to  overcome  the  friction  when 
the  weight  is  doubled,  other  things  being  the  same. 

(5.)  Still  the  time,  during  which  the  two  surfaces  are  acting 
upon  each  other,  either  by  their  own  gravity,  or  by  any  other 
force,  is  to  be  taken  into  the  account,  although  its  effect  has  not 
yet  been  accurately  determined ;  it  is  found  that  the  augmentation 
depending  on  this  cause,  has  its  limits,  and  that  these  limits  vary 
according  to  the  nature  of  the  rubbing  surfaces.  Coulomb  found 
that  in  wood  sliding  on  wood,  without  grease,  the  friction  at  first 
increased,  but  in  a  minute  or  two  came  to  a  limit,  which  it  did 
not  afterwards  exceed.  Oak,  for  example,  sliding  on  oak,  though 
the  pressure  was  varied  from  74"'-  to  2474"'-  had  a  friction  after  a 
minute  always  nearly  44  parts  in  the  hundred. 

In  the  case,  however,  of  iron  rubbing  on  iron,  or  iron  on  brass, 
the  friction  is  the  same  whetiier  the  bodies  are  just  beginning  to 
0)ove  from  rest,  or  have  acquired  any  given  velocity. 
Mech.  18 


138  Statics. 

When  heterogeneous  bodies  are  made  to  slide  upon  one  another, 
as  wood  on  metal,  the  friction  increases  slowly  with  the  time,  and 
does  not  arrive  at  its  maximum  in  less  than  four  or  five  days.  Iron  on 
oak  after  ten  seconds,  is  found  to  have  a  friction  of  7f  per  cent.,  and 
after  four  days  it  amounts  to  nearly  20  per  cent.  In  heterogeneous 
substances,  too,  ihe  friction  increases  sensibly  with  the  velocity,  and 
follows  nearly  an  arithmetical,  while  the  velocity  follows  a  geomet- 
rical progression. 

When  the  surfaces  are  smeared  with  some  unctuous  substance, 
although  the  friction  is  diminished,  a  certain  time  is  required  in  order 
that  the  friction  may  attain  its  maximum.  Oak  rubbing  on  oak,  the 
surfaces  being  covered  with  tallow,  has  a  friction  that  continues  to 
increase  for  five  or  six  days,  and  becomes  stationary  at  about  42  or 
43  per  cent.  In  the  case  of  brass  on  iron  with  fresh  tallow  be- 
tween the  surfaces,  the  friction  is  four  days  in  coming  to  a  maxi- 
mum, when  it  is  10  or  12  per  cent.,  it  being  9  per  cent,  at  the 
commencement.  The  increase  where  metals  are  used,  is  much 
less  considerable  than  in  experiments  with  substances  more  porous 
and  yielding. 

23G.  The  quantity  of  friction  being  determined  for  a  particular 
kind  of  matter,  let  us  now  see  if  the  effect  upon  a  given  machine, 
or  given  motion,  may  be  thence  deduced,  friction  being  considered 
as  simply  proportional  to  the  pressure. 

Fig.  126.  Let  us  take,  as  the  first  example,  the  body  p,  situated  upon  a 
horizontal  plane  AB,  and  drawn  by  the  weight  of  the  body  q, 
parallel  to  AB.  Suppose  that  the  body  q  has  a  weight  just  suffi- 
cient to  put  the  body  p  in  motion.  The  ratio  of  the  weight  q  to 
the  friction  is  thus  found. 

From  the  centre  of  gravity  G  of  the  body  p,  let  fall  the  perpen- 
dicular GH  upon  the  plane  AB.  The  body  p  is  urged  by  gravity 
in  the  direction  GH,  and  by  the  weight  q  in  the  direction  jfiTil/"  which 
meets  GH  in  K.  From  the  joint  action  of  these  two  forces,  there 
will  result  an  effort  according  to  some  line  KI,  meeting  in  1,  the 
horizontal  plane  AB ;  and  this  effort  must  be  just  counterbalanced, 
since  we  have  supposed  that  the  body  p  is  only  upon  the  point  of 
moving.  Suppose  the  effort  according  to  Kl  or  KIZ  applied  at  the 


Friction.  139 

point  /,  and  decomposed  into  two  others,  the  one  perpendicular  to 
the  plane  AB,  and  the  other  in  the  direction  of  this  plane.  These 
efforts  will  evidently  be  the  same  as  those  which  were  directed 
according  lo  KH  and  KL.  IMoreover  ihe  first  will  be  destroyed, 
especially  if  it  meet  the  plane  AB  in  some  point  /  common  to  this 
plane  and  the  surface  of  the  body.  As  to  the  second,  since  it  is  in 
the  direction  of  friction,  it  will  not  be  destroyed  unless  it  happen  to 
be  exactly  equal  to  the  force  of  friction. 

We  hence  see  how  the  value  of  friction  is  to  be  determined  ; 
we  take  successively  for  q  different  weights  until  we  find  one  that  is 
just  sufficient  to  cause  a  motion  in  the  body  7?.  But  not  to  compre- 
hend, in  estimating  the  friction  of  the  body  p,  effects  foreign  to 
that  which  is  sought,  it  is  necessary  to  attend  to  several  particulars  ; 
(1.)  The  pulley  M  should  move  with  die  greatest  ease,  and  the 
cord  KM  q  should  be  as  flexible  as  it  can  be  made.  (2.)  The 
cord  CM  should  be  attached  to  some  point  C  as  near  as  possible  to 
the  plane  AB.  The  necessity  of  this  precaution  arises  from  the 
circumstance,  that  other  tilings  being  the-  same,  the  point  I,  where 
the  effort  in  the  direction  ^/ meets  the  plane  AB,  will  approach  so 
much  the  nearer  to  the  extremity  E  of  the  base  of  the  body,  or  will 
fall  without  the  base  so  much  the  farther  from  the  extremity  E,  ac- 
cording as  the  point  C  is  the  more  elevated  above  the  plane  AB. 
Now  in  the  case  where  the  point  /falls  without  the  base,  the  effort 
perpendicular  to  the  plane  not  being  entirely  destroyed,  there  will 
hence  result  a  tendency  in  the  body  to  rotate  ;  and  the  friction 
thence  arising  would  be  somewhat  more  considerable  than  the 
proper  friction  in  question. 

237.  Let  us  now  consider  a  weighty,  put  upon  an  inclined  plane,  Fig.  127. 
and  retained  by  the  effect  of  friction  alone.  The  action  of  gravity 
directed  according  to  the  vertical  GZ  passing  dirough  the  centre  of 
gravity  G  of  the  body,  and  meeting  in  /some  point  of  the  plane 
AB,  may  be  decomposed  into  two  parts,  one  in  the  direction  of  the 
plane,  and  the  other  perpendicular  to  it.  The  second  will  be  de- 
stroyed, if  the  point  /does  not  fall  without  the  base  DE,  and  the 
first  in  order  to  be  destroyed  must  be  equal  to  the  force  of  friction. 
Now  it  is  evident  that  by  forming  the  parallelogram  IHZL,  IZ 
will  represent  the  weight  of  tlie  body,  IH  the  pressure,  and  HZ  or 


140  Statics. 

IL  the  force  of  friction ;  hence  the  triangles  IHZ,  AB  C  being 
Geom.    similar,  we  have 

202. 

HZ         or         IL  :  IH  ::  BC  :   CA, 

from  which  it  will  be  seen,  that  the  force  of  friction  is  to  the  pressure 
as  the  height  of  the  plane  to  its  base. 

It  will  be  perceived,  in  like  manner,  that 

HZ  :  IZ  .:  BC  :  AB; 

that  is,  the  force   of  friction   is  to  the  weight  of  the  body   as  the 
height  of  the  plane  is  to  its  length. 

In  order,  therefore,  to  determine  the  friction  in  different  sub- 
stances, we  have  only  to  raise  the  plane  AB  till  the  body  p  is  upon 
the  point  of  moving  ;  then  measuring  the  height  and  the  base,  we 
shall  have  the  ratio  of  the   force  of  friction   to  the  pressure. 

238.  By  these  two  examples,  it  will  be  seen,  that   regard   being 
had  to  friction,    the   condition    required  in   order   that  a  body  may 
remain  in  equilibrium  upon  a  proposed  surface,  and   be  in  a  state 
approaching  the  nearest  to  motion,   is,   that  the  single  force   which 
acts  upon  it,  if  there  be  but  one,  or  the  resultant  of  all  the  forces,  if 
there  be  several,  have  with  respect  to  the  surface  upon  which  it  is  to 
move,  an  inclination  GlE  or  LIZ,  such  that  IL  shall  be  to  HI,  as 
the  force  of  friction  is  to  the  pressure.     But  IL  is  to  LZ,  as  one 
Trig.  30. is  to  the  tang.  LIZ,   1  being   the    radius  of   the    tables.     Con- 
sequently the  inclination  LIZ  must  be  such  that  the  radius  shall  be 
to  the  tangent  of  this  inclination,  as  the   force   of  h'iction  is  to  the 
pressure  ;  therefore,  the  ratio  of  the  force  of  friction  to  the  pressure 
being  once  ascertained,  it  will  always  be  easy  to   determine  the   in- 
clination belonging  to  the  resultant  of  all  the  forces  wiiich   act  upon 
the  body,  this  body  being  in  a  state   of  equilibrium   approaching  as 
near  as  possible  to  motion.     Hereafter  we  shall  call  the  angle  LIZ 
the  angle  of  friction.     It  is   different   for   different  substances,   and 
for  different  degrees  of  smoothness  of  the  same   substance.     If  the 
friction  is  33 1  per  cent.,  or  one  third  of  the  pressure,  which  is  near- 
ly the  case  with  respect  to  a  great  many  kinds  of  matter,  the  rubbing 
surfaces  being  tolerably  smooth,  the  tangent  of  LIZ  will  be  triple 
of  the  radius,  that  is,  LZ  will  be  equal  to  3  IL.     Now  the  angle 


Friction.  141 

of  which  the  tangent  is  triple  the  radius  is  71°   34'.     This  will 
accordingly  be  the  angle  of  friction  in  these  cases. 

239.  By  means  of  this  result,  it  will  be  easy  to  deternnine  in 
each  machine  what  ratio  ought  to  exist  between  the  power  and  the 
weight  in  order  that  motion  may  be  upon  the  point  of  taking  place, 
allowance  being  made  for  friction. 

In  the  lever,  for  example,  let  us  suppose  that  the  fulcrum  is  a 
simple  support,  as  represented  in  figure  128.  We  have  seen,  that 
with  respect  to  this  machine,  there  cannot  be  an  equilibrium,  unless 
the  resultant  DF  of  the  two  forces  p,  q,  be  perpendicular  at  F  to 
the  common  tangent  to  the  surface  of  the  lever  and  that  of  the 
fulcrum.  On  the  supposition  of  friction,  the  case  is  different ;  it  163. 
is  still  necessary  that  the  resultant  should  be  directed  from  the  point 
D  to  the  fulcrum  JP;  but  it  is  sufficient  in  order  to  an  equilibrium, 
that  one  of  the  two  inclinations  DFA,  DFB,  according  as  we  wish 
g-  or  p  to  prevail,  should  be  greater  than  the  angle  of  h'iction,  and 
for  the  state  of  equilibrium  approaching  the  nearest  to  motion  on  the 
part  of  the  power  q,  it  suffices  that  the  inclination  DFA  should  be 
precisely  equal  to  the  angle  of  friction  }  since,  if  we  imagine  the 
force  according  to  DF  decomposed  into  two  others,  the  one  perpen- 
dicular to  .,^j5,  and  the  other  in  the  direction  AF,  the  force  in  the 
direction  AF  will  be  less  than  the  friction  in  the  first  case,  and 
exactly  equal  to  it  in  the  second.  With  respect  to  the  two  forces 
q,  p,  they  will  still  be  in  the  inverse  ratio  of  the  perpendiculars 
FK,  FL. 

240.  But  if  the  fulcrum  is  such  that  the  lever  can  have  no  other 
motion  except  fhat  of  a  rotation,  that  is,  if  it  turn  on  an  axis  or  pin, 
we  adopt  the  following  method  which  is  common  to  the  lever,  the 
pulley,  and  the  wheel  and  axle,  especially  when  in  this  last  machine 
the  power  and  weight  are  in  the  same  plane.  We  shall  consider 
first  the  wheel  and  axle  ;  the  manner  of  treating  the  lever  and 
pulley  will  afterwards  appear. 

Let  HCI  be  the  plane  of  the    wheel,   GKL  a  section  of  the  Fig.i29. 
cylinder,  and   NDM  the  axis  about  which  the  machine  is  to  turn. 
On   the  supposition   that  there  is  no  friction,  the   resultant  of  the 
two  powers  y,  q,  passing  through  their  point  of  meeting  jj,  must 
pass  also  through  the  centre  F  of  the  axis.     But  in  the  case  of 


142  Statia. 

friction,  the  machine  must  remain  in  equilibvium  so  long  as  the  di- 
rection of  the  resultant,  supposed  to  be  AD,  does  not  make  with 
the  surface  JVDM  (that  is,  with  the  tangent  at  the  point  uiiere  JID 
meets  this  surface,)  an  angle  less  than  the  angle  of  friction.  This 
will  be  evident  by  imagining  the  force  in  question  decomposed  into 
two  others,  one  perpendicular  to  the  tangent  at  D  ;  and  the  oUier  in 
the  direction  of  this  tangent. 

This  being  done,  since  AD  is  the  directon  of  the  resultant,  we 
48.    shall  have 

q  :  jp  :  :  s\n   GAD  :  sin  DAC, 

:  :  sin  {GAF  +  FAD)  :  sin  {FAC  —  FAD), 

Now,  (1.)  Tf  we  let  fall  upon  AD  the  perpendicular  FE,  in  the 
right-angled  triangle  FED,  the  angle  FDE  is  the  complement  of 
the  angle  EDO  which  AD  makes  at  D  with  the  surface  JVDM, 
and  is  accordingly  supposed  to  be  known;  so  that  if  we  call /the 
angle  of  friction,  FDE  will  be  the  complement  of/,  and  if  we  call 
5  the  distance  FD,  or  the  radius  of  the  axle,  we  shall  have 

Trig. 30.  FE  =  d  cos/ 

the  radius  of  the  tables  being  equal  to  1.  (2.)  As  the  directions  of 
p  and  q  are  supposed  to  be  known,  as  well  as  the  dimensions  of  the 
machine,  the  angles  GAF,  FAC,  are  supposed  to  be  known  as  also 
the  distance  AF.  Thus,  in  the  right-angled  triangle  FAE  in  which 
AF,  FE,  are  known,  it  will  be  easy  to  calculate  the  angle  FAE ; 
Trig. 30. calling  this  angle  e,  and  the  angles  GAF,  FAC,  a,  h,  respectively, 
48-    we  shall  have 

q  '.  p  :  '.  swx  {a  -{•  e)  :  sin  [h  —  e) ; 

consequently  q  = r^ — *"    ' ;  this  is  the  value  of  the  power 

in  the  case  of  friction. 

If  the  friction  is  nothing,  the  angle /of  the  friction  is  90°;  that 
is,  the  resultant  must  be  perpendicular  to  the  surface  of  the  axis, 
and  consequently  pass  through  the  centre  F.  We  have  accordingly, 
cos/  =  0,  and  hence  FE  =  0,  and  e  =  0 ;  therefore 

p  sin  a 

"  sin  6   ' 


Friction.  143 

Now  regarding  FA  as  radius,  the  perpendiculars  FG,  FC,  are 
the  sines  of  the  angles  FAG,  FAC,  or  of  a,  6  j  we  have,  there- 
fore, hy  calling  FG,  D,  and  FC,  D', 


whence 


and 


J)  :  D'  :  :  sin  a  :  sin  h, 


D 


sin  b         D' 


q  =  -^,    org:p::D:D', 


which  agrees  with  what  has  already  been  proved.  178. 

If  the   directions  p  A,  q  A,  are   parallel,  the   angles  GAF, 
FAE,  FAC,  are  considered  as  infinitely  small,  and  consequently  as 
having  die  same  ratio  as  their  sines.     We  may  accordingly  sub- Trig.  17. 
stituie  sin  a  -j-  sin  e  for  sin  (a  -j-  e),  and  sin  b  —  sin  e  for 

sin  (b  —  e), 
and  we  shall  have 

p  (sin  a-|-sin  e) 
■*  sm  0  —  sin  e 

But  we  have  just  seen,  that 

D  :  D'  :  :  sin  a  :  sin  b, 

and  for  the  same  reason, 

sin  a  :  sin  e  :  :  FG  :  FE, 

;  :     D    '.  d  cos/ J 


whence 


and 


D  sin  b 
sin  a  = jj, — , 


d  cos  f  s\n  a            S  cos  ^sin  6 
s.n  e  =  j^ =  jy, ; 

substituting  in  the  value  of  q,  for  sin  a  and  sin  e  the  values  above 
found,  we  have 


144  Statics. 

/D  sin  b     ^^  S  cosy  sin  b\ 

__P  \U  ^  D         )         p  {D  +  8cosf) 

.     ,         d  cos  /'sin  b  X>'  —  d  cos  /' 

sin  6 jj. "^ 

Therefore,  since  when  there  is  no  friction,  the  value  of  the  power  is 

p  D  . 

^-jy,  if  we  call  z  the  augmentation  which  the  power   must   receive 

on  account  of  friction,  we  shall  have 

_p  {D  +  S  cos/')        pD 
^~     D'—dcosf       ~Iy' 

_p  D'  (D  -x-  8  cos  f)—pDD'  -\- p  D  8  cos/ 
~  D'  {D'  —  (TcoT/)  ' 

_p{D'  -\-  D)  8  cnsf  ^ 
~  ~D  {D'  —  8  cos  /)  ' 

from  which  it  will  be  seen,  that  the  effect  of  friction  will  be  less 
according  as  the  radius  of  the  axle  is  less,  although  it  does  not 
diminish  exactly  in  proportion  to  this  radius. 

241.  This  solution  adapts  itself  to  the  lever,  by  regarding  D^ 
and  D  as  the  distances  of  the  directions  of  the  two  forces  from  the 
fulcrum.  It  is  applicable  also  to  the  fixed  pulley,  by  supposing 
jy  =:  D,  which  would  give, 

2  p  5  cos  /" 


D  —  8  cosy 

Although  in  the  wheel  and  axle  we  have  supposed  the  direc- 
tions in  the  same  plane,  the  solution  is  not  the  less  adapted  to  the 
ordinary  construction  of  this  machine  in  which  the  weight  and  pow- 
er are  exerted  in  planes  but  litde  distant  from  each  other. 

242.  To  ascertain  the  effect  of  friction  in  the  movable  pulley 
Fig.130.  we  proceed  thus.  In  order  that  the  power  q  may  be  upon  the 
point  of  causing  the  pulley  to  turn  about  its  axis  i^,  it  is  necessary 
that  it  should  receive  an  augmentation  sufficient  to  overcome  the 
friction.  Now  by  this  augmentation,  the  power  causes  the  weighty 
to  depart  from  its  position  to  such  a  degree,    that  a   vertical    drawn 


Friction.  145 

through  the  centre  of  gravity  shall  make  at  the  point  D,  with 
the  surface  of  the  axle,  an  angle  equal  to  that  of  friction. 
Then   any  increase  of  the  power  will  cause  tiie   pulley  to   turn. 

By  the  position  which  the  weight  has  taken,  it  will  tend  to 
turn  the  axle  upon  its  centre  P,  with  a  force,  the  moment  of  which 
will  be  p  X  FE,  FE  being  perpendicular  to  a  vertical  passing 
through  D.     Now  we  have  already  seen  that 

FE  —  d  cos  f;  240. 

this  moment,  therefore,  will  he  p  8  cos  /.  But  in  order  that  the 
power,  by  its  augmentation,  which  I  shall  call  r',  may  be  upon  the 
point  of  overcoming  this  effort,  it  is  necessary  that  the  moment 
z'  X  FG  of  the  force  z',  with  which  it  tends  to  cause  a  rotation 
about  F,  should  be  equal  to  the  moment  p  8  cos/j  we  have,  there-  172. 
fore,  by  calling  the  radius  FG,  D, 

D  z'  =  p  d  cos  f, 

and  consequendy, 

_  pdcosf ^ 


D 


y 


from  which  it  will  be  seen,  that  the  effect  of  friction  will  be  less 
according  as  the  radius  of  the  axle  is  less  than  that  of  the  pulley, 
and  in  the  same  ratio. 

It  will  hence  be  easy  to  determine  the  effect  of  friction  in  the 
different  systems  and  combinations  of  pulleys.  Suppose  that  the 
question  related  to  one  similar  to  that  represented  in  figure  92,  the 
weight  p  being  equal  to  400"^-,  and  the  radius  8  of  the  axle  as  well 
for  the  fixed  as  for  the  movable  pulleys  being  a  fifth  part  of  the 
radius  D  of  the  pulley. 

The  two  cords  1 ,  2,  sustain  half  of  the  weight  or  200"'- ; 
thus,  if  in  the  value  of  z,  we  put  for  p  SOO""-,  \  D  for  d,  and 
on  the  supposition  that  the  friction  is  one  fourth  of  the  pressure, 
which  would  give  the  angle  of  friction  equal  to  75°  58',  and  the 
cos  /  or  cos  75°  58,  equal  to  0,24249,*  or  simply  0,24,  we  shall 
have, 

*  See  table  of  natural  sines  and  cosines,  Topography. 
Mech.  19 


146  Statics. 

^  jJ  d  COS  f 

~  —      />    ' 

=  200  X  i  X  0,24  —  9"'-,  6, 

which  is  an  expression  for  the  excess  of  tension  of  the  cord  2  over 
the  cord  1,  on  account  of  friction. 

Now  the  cord  2  and  the  cord  3  passing  over   the  fixed  hlock, 
we  shall  have   the  excess  of  the   tension  of  the  cord   3,  over  the 
241.        cord  2,  by  the  formula, 

2  p  d  cos  / 

^  ~~  D  —  d  cos/' 

in  which  p  represents  the  tension  of  the  cord  2,  or  109"'-,  6  ;  since, 
without  friction,  this  tension  would  be  one  fourth  of.  the  weight  or 
lOO"'-.     Accordingly  we  have 

219.2X1X0,24    _  10,522  _       , 
^-       i_o,24xi       ~    0,95     ~         ' 

This  is  the  quantity  by  which  the  cord  3  is  more  stretched  than  the 
cord  2,  on  account  of  friction.  Thus  the  cord  3  is  stretched  with  a 
force  equal  to  120i''-  7. 

As  the  cords  3  and  4  embrace  the  movable  block,  we  deter- 
mine how  much  the  cord  4  is  more  stretched  than  the  cord  3  by 
the  formula 

,,  _  P  Scos  f 
-   -         jj        , 

which  would  give  as  above  z'  =  O"'-,  6.  Thus  the  cord  4  is 
stretched  with  a  force  equal  to 

100  +  9,  6  4-  11,  1  +  9,  G  =  ISO'b-,  3. 

Finally,  by  supposing  for  greater  simplicity,  which  will  make 
but  little  difference  in  the  result,  that  the  two  cords  4,  5,  which 
embrace  the  fixed  block,  are  parallel,  we  shall  have  the  quantity 
by  which  the  cord  5  is  more  stretched  than  the  pord  4  by  the 
formula 

2  p  5  cos  / 

"  ""  D—dcosf  ' 

which  gives 


Friction.  147 

^  -_  260,0  xjX  0,24  _      ,b.  g 

Thus  the  tension  of  the  cord  5,   which  without  friction  would  be 
only  100,  is  equal  to  130,  3  +  13,  2  =  143"^-,  5. 

243.  In  the  determination,  which  we  have  given,  of  the  effect 
of  friction  upon  the  movable  pulley,  we  have  not  allowed  for 
any    increased    pressure    at  D   arising   from    the   augmentation   of 

the  power,  although  this  is  done  by  some  writers.  The  reason  Fig.  130. 
is,  that  this  augmentation  of  the  power  contributes  nothing  to  the 
pressure  at  D ;  indeed,  setting  aside  the  stiffness  of  the  cords,  and 
certain  odier  obstacles,  whenever  the  power  is  greater  than  is 
necessary  to  an  equilibrium,  the  body  of  the  pulley  is  elevated 
by  this  excess.  The  augmentation  of  the  power  does  not  con- 
tribute any  thing  to  the  friction  against  the  axis,  if  no  regard  is 
had  to  the  velocity  that  may  thus  be  produced,  which  together  with 
the  inertia  of  the  weight  and  pulley  are  at  present  left  out  of 
consideration.  It  is  only  the  weight  which  presses.  This  is  not 
the  case  with  the  fixed  pulley  ;  and  the  method  which  we  have 
given  comprehends,  with  respect  to  this  point,  every  thing  which 
ought  to  be  taken  into  the  account,  although  it  differs  from  the 
course  heretofore  adopted,  in  which  something  has  been  taken  for 
granted. 

We  will  not  deny,  that  in  the  calculation  which  we  have  given 
of  the  effect  of  friction,  if  we  would  know  the  effect  of  friction 
very  rigorously,  it  would  be  necessary  to  consider  the  subject  aF'g-92. 
little  differently.  In  fact,  by  determining  in  the  way  we  have  done, 
the  particular  tensions  of  each  of  the  cords,  it  is  taken  for  granted 
that  each  pair  of  cords  acts  as  it  would  do  in  the  case  of  a  simple 
pulley,  which  is  not  strictly  true,  perhaps.  But  this  approximation 
will  suffice  for  the  present. 

244.  In  order  to  determine  upon  the  inclined  plane,  the  ratio 
of  the  power  to  the  weight,  when  the  former  is  upon  the  point  of 
causing  the  body  to  move,  we  proceed    thus.     Through  the    point 

of  meeting  i*^,  of  the  directions  of  the  power   </,  and    weighty,    weFig.131. 
suppose  the  line  Fl  drawn,  making  with    the    plane   AB   an  angle 
FIA  equal  to  that  of  friction.     In  order  that  the  power   q   may   be 
upon  the  point  of  causing  motion,   it  is   necessary,  (1.)   That   the 


Geom. 


148  Statics. 

resultant  of  the  power  and  weight  should  be  directed  according 
to  FI.  (2.)  That  the  point  /,  where  the  line  FI  meets  the  plane, 
should  belong  to  some  point  of  the  base  DE  j  otherw'ise  the  body 
would  tend  to  turn. 

This  being  premised,  we  have 

p  :  q   :  :  sin  q  FI  :  sin  p  FI, 

or,  letting  fall  upon  the  plane  the  perpendicular  FH, 

p  :  q  ::  sin  {q  FH—  HFI)   :  sin  (p  FH  +  HFI). 

Now  the  angle  HFI  is  the  complement  of  the  angle  of  friction ; 
and  the  angles  q  FH,  p  FH,  are  supposed  to  be  known,  since  the 
direction  of  the  power  is  considered  as  known,  together  with  the 
209.  inclination  of  the  plane,  which  is  equal  to  the  angle  p  FH;  we 
have  accordingly  the  ratio  of  p  to  q. 

If  we  would  determine  this  ratio  in  lines,  we  have  only  to 
draw  through  any  point  B  of  the  inclined  plane,  the  line  BT, 
making  with  AB  the  angle  ABT  =  HF  q,  and  the  line  BV,  mak- 
ing with  AB  the  angle  ABV  =  HFI,  the  complement  of  the 
angle  of  friction.  Then  drawing  the  horizontal  line  AT,  we  shall 
have 

p  :  q  :  :   VT  :  BT, 

Geom.    since  the  angle  VBT  =  ABT  —  ABV  =  HF q  —  HFI,  and 
'^^'        the  angle  BVT  =  BAV  +  ABV  =  p  FH  +  HFI.     Now  in 

the  triangle  BVT 

Tn6.32.  VT  :  BT  ::  sin  VBT  :  sin  BVT. 

Instead  of  making  the  angle  ABT  =  HF  q,  and  the  angle 
ABV  =  HFI,  we  may  draw  BT  perpendicular  to  the  direction 
of  the  power  q,  and  BV  perpendicular  to  FI;  this  amounts  to 
the  same  thing,  and  is  moreover  analogous  to  the  course  pursued 
in  article  203. 

245.  The  second  condition  shown  to  be  necessary  in  order 
that  the  power  q  may  be  upon  the  point  of  moving  the  body, 
renders  it  evident,  that  when  the  body  does  not  rest  upon  a  point, 
the  direction  of  the  power  produced  must  meet  the  vertical. 
Fig.  132.  drawn  through  the  centre  of  gravity,  at  the  point  JP,  where  this 
last  line  is  met  by  the  line  IF,  proceeding  from  some  point  of  con- 


Friction.  149 

tact  1,  and  making  with  the  plane  an  angle  equal  to  the  angle  of 
friction. 

246.  We  proceed  in  the  same  manner  in  determining  the 
second  kind  of  friction,  or  that  which  is  to  be  overcome  in  giving 
a  rolling  motion  to  bodies  terminated  by  curved  surfaces ;  I  say 
curved  surfaces,  since  with  respect  to  bodies  terminated  by 
plane  surfaces,  as  they  cannot  roll,  except  by  turning  on  a  point 
or  some  angular  part,  we  shall  not  treat  of  these,  the  laws  and  value 
of  friction  in  such  cases  not  being  sufficiently  known.  But  as  to 
the  friction  of  bodies  terminated  by  curved  surfaces,  the  method 
is  precisely  the  same  as  that  above  pursued.  We  have  only  to 
"suppose  the  angle  of  friciion  to  approach  more  nearly  to  90°  ;  and 
it  is  to  experiment  that  we  are  to  look  for  the  determination  of  this 
angle  in  all  cases. 

247.  Friction  may  be  the  occasion  of  motions  very  different 
from  those  which  take  place  without  this  cause,  some  of  which  it  may 
be  worth  while  to  notice. 

We  have  already  mentioned  more  than  once,  what  must  hap- 136,  &c. 
pen  to  a  free  body  50  5',  which  receives  an  impulse  in  a  direc-Fig.lss. 
tion  not  passing  through  the  centre  of  gravity.  But  if  the  body 
were  struck  externally  according  to  any  direction  AB,  it  would 
not  receive  the  whole  of  this  impulse.  The  impelling  force  is 
to  be  decomposed  into  two  others,  one  in  the  direction  of  a  tan- 
gent to  the  surface,  and  the  other  perpendicular  to  this  surface. 
When  there  is  no  friction  the  impelling  force  would  have  no 
effect  in  the  direction  of  a  tangent.  It  is  only  the  force  in  the 
direction  BF,  therefore,  which  would  be  transmitted  to  the  body, 
and  this  would  not  cause  the  body  to  turn,  except  when  it  hap- 
pened not  to  pass  through  the  centre  of  gravity  G.  It  will  hence 
be  seen  that  if  the  body  were  of  a  spherical  form  and  homogene- 
ous, it  would  never  be  made  to  turn  in  virtue  of  an  external  force 
unaccompanied  with  friction,  since  in  this  case  a  perpendicular 
to  the  surface  would  always  pass  through  the  centre  of  the  figure, 
which  would,  at  the  same  time,  be  the  centre  of  gravity.  On  the 
supposition  of  friction,  the  case  is  different.  The  force  in  the 
direction  of  a  tangent  would  transmit  itself  by  means  of  the  as- 
perities of  the  surface,  and  to  a  greater  or  less  degree  according 
to  the  amount  of  friction  ;  so  that  in  addition  to  the  motion  aris- 


150  Statics. 

ing  from  the  perpendicular  force  BF,  the  body  would  turn,  and 
the  centre  of  gravity  G  would  advance  in  a  line  parallel  to  the 
tangent,  as  if  the  point  B  were  drawn  in  tiiat  direction  by  means 
of  a  thread  attached  at  this  point,  and  with  a  power  equal  to  the 
force  of  friction. 

Fig.  134.  24S.  Let  us  suppose  that  a  hard,  spherical  body  ABC,  falls 
freely  upon  a  horizontal  plane  HR,  and  that  it  receives,  from  some 
cause  or  other,  a  motion  of  rotation  about  its  centre  of  gravity  ; 
if  there  were  no  friction,  this  body,  after  meeting  the  plane  would 
preserve  only  its  rotatory  motion,  and  its  centre  of  gravity  would 
be  at  rest.  But  in  the  case  of  friction,  when  the  body  lias  reached 
the  plane,  it  will  roll  from  /toward  B,  or  from  /toward  //,  accord- 
ing as  the  rotatory  motion  is  in  the  direction  CAB,  or  in  that  of 
BAG)  since  the  resistance  of  friction,  which  is  exerted  in  the  direc- 
tion of  the  plane,  is  equivalent  to  a  force  acting  upon  this  body  in 
a  direction  opposite  to  its  motion ;  and  as  it  does  not  pas3  through 
the  centre  of  gravity  of  the  body,  it  must  give  it  a  motion  parallel 
to  the  plane,  and  a  rotatory  motion,  both  in  the  direction  contrary 
136.  to  the  actual  motion  of  rotation.  Now  of  these  two  motions,  the 
latter  diminishes  continually  the  original  motion  ;  and  on  the  other 
hand  the  motion  of  the  centre  will  be  accelerated  to  a  certain  point, 
after  which  it  will  be  diminished  till  it  is  destroyed  with  the  motion 
of  rotation. 

249.  We  are  hence  enabled  to  explain  several  phenomena ; 
Flg.135. as  (1.)  Why  a  spherical  body  ABC,  struck  in  the  direction  DB, 
after  having  advanced  In  the  direction  IE,  retiuns  afterward 
from  E  toward  /,  and  may  even  pass  beyond  /  toward  H.  The 
impulse  in  the  direction  BB  causes  it  to  turn  (on  account  of  fric- 
tion at  B),  according  to  ABC,  and  to  advance  in  the  line  IE',  but 
the  friction  upon  the  plane,  being  now  a  friction  of  the  first  kind,  the 
motion  of  the  centre  of  gravity  is  soon  destroyed,  and  the  motion 
of  rotation  gives  rise  to  another  in  the  opposite  direction,  as  in  the 
248.    preceding  case. 

(2.)  We  are  moreover  furnished,  upon  the  same  principles, 
with  the  reason  why  a  cannon-ball,  which  had  apparently  lost 
nearly  all  its  force,  seems  on  striking  to  recover  it  again,  and 
often  with  violence.     When  it  is  impelled  by  the   force   of   the 


Friction.  151 

powder,  it  acquires  at  the  same  time,  in  consequence  of  the 
friction  against  the  bottom  of  the  bore,  a  rotatory  motion,  which 
is  but  little  affected  while  in  the  air ;  but  when  the  ball  comes  to 
reach  the  ground,  as  the  rotatory  motion  on  the  part  toward  the 
ground  takes  place  in  a  direction  opposite  to  the  progressive  mo- 
tion, the  consequence  must  be  an  acceleration  in  the  motion  of  the 
centre,  that  is,  of  the  progressive  motion.  248. 

250.  Finally,  if  friction  is  disadvantageous  in  many  cases,  it 
is  still  not  without  its  utility.  Were  it  not  for  this,  the  least  in- 
clination would  be  continually  subjecting   us  to  a  fall.      No   man 

or  other  animal  could   turn   while   in  rapid  motion  without  falling, 

...  .,,  ,  c  f     ■        Fig.136. 

whatever  position  he  might  take  ;  whereas,  on   account  ol  tnction, 

an  animal  may  incline  himself  toward  the  point  jP,  for  example, 
about  which  he  is  moving,  in  such  a  manner  that  his  weight,  di- 
rected according  to  the  vertical  GK,  passing  through  the  centre 
of  gravity  G,  and  the  tendency  to  fly  off  GC,  acquired  by  turn- 
ing, and  which  is  directed  from  F  toward  C,  will  conspire  to  pro- 
duce a  single  force  according  to  the  line  GI,  passing  through  a 
point  I  between  the  legs  of  the  animal ;  this  force,  although  ob- 
lique, is  still  destroyed  by  friction,  provided  the  inclination  be 
within  the  limits  required  by  the  laws  of  friction. 

It  is  moreover  to  friction  that  we  are  indebted  for  the  power 
of  diminishing  friction,  when  injurious;  since  it  is  only  by  means 
of  this  resistance  that  we  are  able  to  work  and  polish  the  surfa- 
ces of  bodies.  It  is  to  friction  that  we  owe  the  facility  with  which 
the  parts  of  certain  machines  are  rendered  sometimes  fixed,  and 
sometimes  movable.  It  is  by  friction  that  scissors  and  the  like 
Instruments,  pincers,  forceps,  files,  &tc.,  produce  their  effect.  If  the 
parts  of  scissors,  for  example,  were  not  saws,  armed  with  small 
teeth,  which  take  into  the  cavities  of  the  bodies  to  be  cut,  these 
bodies  would  slip  from  between  the  two  edges. 

Friction  is  also  very  often  of  service  in  moving  bodies  in 
certain  directions  ;  thus  if  we  would  raise  the  body  p  by  means 
of  the  lever  AB,  it  is  very  easily  done,  by  making  the  body  bear 
on  the  edge  CD.  The  friction  in  a  case  like  this,  being  veryFig.137. 
considerable,  renders  CD  fixed,  and  prevents  the  body  from  slip- 
ping.   The  same  cause  keeps  the  extremity  A  of  the  lever  in  its 


1 52  Statics. 

place.  In  this  case,  if  we  would  know  the  ratio  of  the  weight  p 
to  the  power  q,  we  imagine  the  weight  of  p,  directed  according^ 
to  the  vertical  GK,  passing  through  the  centre  of  gravity  G,  de- 
composed into  two  parallel  forces,  the  one  passing  through  the 
point  /  in  which  the  body  rests  upon  the  lever,  and  the  other 
through  a  point  of  CD,  situated  in  the  plane  of  the  two  parallels 
52.  GK,  IM;  then  the  resulting  force  in  1  will  be  tojt?  as  EK lo  EM; 
and  if  from  A  we  let  fall  upon  IM  the  perpendicular  AL,  the 
force  q  will  be  to  the  force  at  /,  as  AL  to  AB  ;  whence 

q  :  p  ::  AL  X  EK  :  AB  X  EM. 

In  short,  it  is  only  on  account  of  the  friction  at  the  point  /  that 
we  regard  the  force  according  to  IM,  as  transmitted  entirely  to 
the  lever.  The  lever  would  otherwise  receive  only  a  part  of 
this  force  which  would  exert  itself  in  the  direction  of  a  perpen- 
dicular to  AB. 

251.  It  is  to  friction  and  friction  only,  that  we  are  to  refer 
the  singular  motion  by  which  certain  bodies  in  a  state  of  rotation 
are  seen  to  elevate  themselves  contrary  to  the  tendency  of  grav- 
ity ;  I  speak  of  the  phenomena  of  the  top.  It  is  well  known  that 
when  a  body  of  this  description,  that  is,  one  which  is  symmetri- 
Fig.i38.cal  with  respect  to  one  of  its  axes,  as  ND,  has  received  a  rotato- 
ry motion  about  this  axis,  and  moves  upon  its  point  N,  over  a 
horizontal  plane  XZ;  it  is  well  known,  I  say,  that  the  smaller 
the  point  iV,  and  the  greater  the  divergency  of  the  sides  from  it, 
the  greater  is  the  tendency  of  this  body  to  rise,  and  thus  to  place 
the  axis  ND  in  a  vertical  position.  We  proceed  now  to  show 
that  this  phenomenon  could  not  take  place  without  friction ;  we 
shall  speak,  moreover,  of  the  nature  of  this  friction. 

Fig.139.  To  simplify  the  subject,  let  us  consider  only  the  axis  ND  of 
the  top,  and  let  us  suppose  the  point  N,  and  the  horizontal  plane 
XZ,  to  be  perfectly  smooth.  The  only  cause  which  opposes  the 
motion  of  the  centre  of  gravity  G,  being  the  plane  XZ,  the  resis- 
tance which  the  centre  of  gravity  meets  with  can  have  no  other 
direction  than  the  line  NK,  perpendicular  to  XZ,  whatever  be 
the  rotatory  motion  about  the  axis  ND.  Now  it  is  evident  that 
this  resistance  takes  place   only  because  gravity  urges  the  body 


Friction.  153 

towards  the  plane,  for  the  rotatory  motion  about  the  axis  ND 
cannot  cause  any  pressure  upon  this  plane ;  the  resistance  in 
question,  therefore,  can  never  be  equivalent  to  the  force  of  2;rav- 
ity,  and  there  will  always  remain  in  the  centre  of  gravity  G,  a 
force  tending  to  bring  it  to  the  plane.  Hence,  when  there  is  no 
friction,  and  the  top  has  received  at  first  no  motion  of  rotation, 
except  about  the  axis  of  its  figure,  it  must  of  necessity  fall. 

252.  It  is  not  the  same  on  the  supposition  of  friction.  In 
this  case,  the  resistance  which  takes  place  at  iV^  acts  not  accord- 
ing to  the  perpendicular  NK,  but  according  to  the  line  NK', 
which  makes  with  the  phme  XZ  an  angle  equal  to  the  angle  of  fric- 
tion, and  passes  through  N,  one  of  the  points  of  friction.  What- 
ever may  be  this  angle  and  this  point,  the  resistance  which  takes 
place  along  the  line  NJC,  is  equivalent  to  a  force  acting  upon 
the  body  in  a  contrary  direction ;  now,  as  this  direction  does  not 
pass  through  the  centre  of  gravity,  it  must  produce  in  the  body 
a  rotatory  motion,  that  is,  a  variation  in  its  actual  motion  of  ro-  137. 
tation  ;  but  it  must  also  transmit  itself  entirely  to  the  centre  of 
gravity.  Let  us  suppose,  therefore,  that  GL  parallel  to  NK'  is 
this  force ;  if  the  vertical  line  GI  represent  the  force  of  gravity, 
and  the  parallelogram  GLEIhe  completed,  GE  will  be  the  actual 
force  which  belongs  to  the  centre  of  gravity  G. 

Now,  the  angle  LGI  and  the  force  GI  remaining  the  same, 
the  greater  the  force  acting  in  the  direction  K'JSf,  and  consequent- 
ly the  greater  the  force  GL,  the  more  nearly  will  the  line  GE 
approach  the  line  GL ;  that  is,  the  greater  will  be  the  tendency 
of  the  point  E  to  rise  above  G.  It  remains  therefore  to  be  seen, 
whether  from  the  nature  of  friction,  together  with  the  figure  of 
the  body,  and  its  motion  of  rotation,  the  ratio  of  the  force  in  the 
direction  JV^'  (or  the  force  GL)  to  the  force  of  gravity  GI,  canFig.138. 
be  increased  till  the  point  E  shall  be  above  the  point  G ;  in 
which  case  it  is  clear  that  the  centre  of  gravity  may  rise  with 
respect  to  the  plane  ;  yet  not  so  as  to  cause  the  point  N  to  quit 
it,  because  the  motion  of  rotation  which  results  from  the  force 
in  the  direction  K'N,  will  tend  to  bring  this  point  toward  the 
plane.  Now  (I.)  As  the  body  rests  upon  a  point,  it  cannot  be 
denied  that  the  parts  of  this  point  sink  more  deeply  than  if  the 
body  rested  upon  a  sensible  surface.  Regard  being  had  to  the 
rotatory  motion  about  ND,  and  to  the  action  of  gravity,  the  pres- 

Mech.  20 


154  Statics. 

sure  exerted  upon  JV  is  by  no  means  the  effect  of  gravity  only. 
To  have  a  just  idea  of  this  pressure,  we  must  consider  that  by 
gravity  the  parts  of  the  point  N  are  at  first  urged  against  the 
plane ;  (2.)  That  by  friction  they  are  kept  there  with  a  certain 
degree  of  force  ;  (3.)  That  by  the  rotatory  motion  they  tend  to 
penetrate  still  further  into  the  plane  ;  of  this  we  shall  be  con- 
vinced by  observing  how  readily  instruments  designed  to  pierce  by 
turning,  are  made  to  penetrate,  when  once  introduced  by  means 
of  an  opening  however  slight.  Now  all  this  is  strictly  applicable 
to  the  top ;  the  parts  of  the  point  are  attached  by  friction  ;  and 
in  this  way  the  rotatory  motion  is  aided  in  effecting  an  entrance 
into  the  plane.  This  motion,  moreover,  being  the  more  rapid 
and  the  better  fitted  to  bore  and  ()ress  upon  the  surface  XZ.  ac- 
cording as  the  parts  of  the  body  diverge  more  from  the  axis  ND 
in  receding  from  the  point  N,  it  must  produce  the  effect  which 
takes  place  in  the  instruments  abovementioned,  that  is,  it  must 
urge  forward  so  much  the  more  forcibly  the  parts  of  the  point  in 
question.  From  all  this,  it  is  evident  that  the  more  the  figure  of 
the  top  diverges  from  the  point  N,  and  the  more  rapid  the  rota- 
tory motion,  the  greater  will  be  the  force  GL  com[)ared  with  grav- 
ity, and  consequently  the  more  will  the  resultant  GE  tend  to  raise 
the  centre  of  gravity  above  the  plane.  Now  it  is  clear  that  in 
proportion  as  the  force  by  which  the  point  is  supported,  and 
at  the  same  time  the  tendency  of  the  centre  to  rise,  become  more 
considerable,  by  so  much  will  the  tendency  of  the  axis  ND  to- 
ward a  perpendicular  to  the  plane  be  increased  ;  so  that  when 
ND  becomes  vertical,  it  begins  after  a  time,  to  incline  more  and 
more,  and  if  the  inequalities  of  the  surface  are  not  too  great,  the 
top  will  be  seen  to  rise  above  the  plane  in  very  small,  sudden, 
vertical  leaps  ;  and  this  we  in  fact  observe  when  the  point  ter- 
minates in  a  small  plane  surface  cut  very  square  and  perpendicular 
to  the  axis. 

We  have  sensible  proof  of  the  truth  of  this  explanation  deriv- 
ed from  the  fact,  that  the  pressure  of  the  point  upon  the  plane, 
is  much  greater  than  that  arising  from  gravity  simply.  Indeed, 
when  the  top  is  put  in  motion  upon  a  plane  of  yielding  matter, 
the  point  works  its  way  into  the  substance  of  the  plane  ;  and  if  it 
be  taken  into  the  hand,  the  pressure  will  become  much  more 
sensible  than  that  which  takes   place  when  there  is  no  motion. 


Stiffness  of  Cords.  155 

We  learn  at  the  same  time  from  this  experiment,  that  the 
phenomenon  in  question  requires  (1.)  That  the  point  should  be 
small  compared  with  the  distance  of  the  parts  of  the  top  from  the 
axis  ND  ;  and  (2.)  That  these  parts  should  turn  wiih  consider- 
able rapidity  ;  and  the  success  will  be  more  or  less  complete,  as 
these  conditions  are  more  or  less  perfectly  lulfilled. 

It  will  be  seen,  moreover,  that  upon  an  inclined  plane  the  top 
must  have  a  tendency  not  to  a  vertical  but  to  a  perpendicular 
to  the  plane.  But  as  it  must  at  the  same  time  slide  along  the 
plane,  and  as  this  motion  would  cause  a  great  vacillation  in 
passing  over  the  inequalities  of  the  plane,  it  will  not  so  easily 
preserve  its  perpendicular  position  as  if  the  plane  were  hori- 
zontal. 


Of  the  Stiffness  of  Cords, 

253.  The  stiffness  of  ropes  and  cords,  or  the  difficulty  withFig.l40. 
which  they  are  bent  into  a  given  curve,  is   also  one  of  the  causes 
which  diminish  the  effect  of  forces  applied  to  machines. 

In  order  to  understand  in  what  manner  this  stiffness  impairs 
the  effect  of  forces,  let  us  suppose  the  wheel  or  pulley  ABC  to  be 
movable  about  the  axle  /.  without  friction.  The  two  weights  p 
and  q  being  equal,  if  we  make  a  very  small  addition  to  one  of  them, 
as  q,  for  example,  no  motion  will  follow,  unless  the  cord  p  ABC  q 
be  perfectly  flexible.  Indeed,  if  we  imagine  that  this  cord, 
instead  of  being  perfectly  flexible,  is  perfecily  inflexible,  so  that 
the  parts  A  p,  C  q,  are  stiff  rods  firmly  fixed  to  the  body  of  the 
pulley  ;  it  is  evident  that  the  pulley  being  moved  by  an  external 
force  in  the  direction  ABC,  the  two  weights  p  and  q  will  take  the 
situations  p'  and  q^ ;  but  they  will  tend  to  return  to  their  first 
position,  and  can  be  prevented  only  by  the  constant  exertion 
of  a  particular  force.  If,  then,  the  cord  is  neither  perfectly  in- 
flexible, nor  perfectly  flexible,  the  effect  of  this  imperfect  flexi- 
bility will  be,  that  the  point  A  passing  to  A',  and  the  point  C  tOFig.i4i. 
C,  the  parts  A'  p',  C  q',  will  be  a  litfle  bent,  and  in  such  a  man- 
ner that  the  weight  p'  will  be  farther  from  /,  and  the  weight  q' 
nearer  to  it,  than  they  would  be  if  the  cord  were  perfectly  flexi- 


15G  Statics. 

ble  ;  so  that  a  certain  force  is  required  in  order  to  bring  the  parts 
A'O,  CC,  into  the  direction  of  tangents  to  the  points -4  and  C; 
in  other  words,  a  force  must  be  employed  which  would  be  un- 
necessary but  for  this  want  of  flexibility. 

The  pulley  being  always  supposed  to  move  with  perfect  ease 
upon  its  axle  J,  if  instead  of  a  cord  a  ribbon  be  employed,  a  very 
small  increase  in  the  weight  q  will  cause  the  pulley  to  turn.  But 
if  the  cord  be  replaced,  it  will  evidendy  be  necessary  to  augment 
the  weight  q  ;  (1.)  According  as  the  sum  of  the  weights  p  and  q, 
or,  in  general,  the  whole  force  by  which  the  cord  is  stretched,  is 
more  considerable  ;  because,  other  things  being  the  same,  the  re- 
sistance occasioned  by  the  weights  p  and  q,  when  by  the  stiffness  of 
the  cord  they  take  the  positions  A'O  p',  CO  q',  will  increase  as 
the  weights  themselves  increase. 

(2.)  The  addition  to  be  made  to  q,  must  be  greater  according 
as  the  radius  of  the  pulley  (or  of  the  surface  over  which  the 
cord  passes)  is  less.  For  the  resistance  which  the  power  meets 
with  arises  from  this,  that  the  cord,  instead  of  adapting  itself  to 
the  revolving  surface,  remains  at  a  certain  distance,  forming  a 
curve  p'  OA'  and  making  with  the  surface  an  angle  OA'A ;  and 
this  resistance  will  evidently  be  the  greater,  according  as  the 
curvature  A'O  of  the  cord  departs  more  from  the  curvature  of  the 
surface  ;  that  is,  according  to  the  smallness  of  the  radius  of  this 
surface. 

(3.)  The  power  applied  must  also  be  increased  in  proportion 
to  the  diameter  of  the  cord.  Indeed  it  is  manifest,  that,  other 
things  being  the  sajne, -the  cord  will  bend  the  less  according  as  the 
thickness  is  greater ;  but  we  have  just  seen  that  the  resistance  to 
the  power  is  greater  according  as  the  curve  A'O  differs  more 
from  the  curve  A' A  ;  it  is  therefore  the  greater  according  as  A'O 
differs  less  from  a  straight  line,  or  the  position  of  an  inflexible 
rod,  that  is,  according  as  the  cord  has  a  greater  diameter  or 
radius. 

254.  Let  us  suppose  that  k  is  the  addition  to  be  made  to  a 
power  to  render  it  sufficient  to  overcome  the  resistance  arising 
from  the  stiffness  of  the  cords,  when  the  entire  force  by  which 
the  cord  is  stretched  is  p,  the  diameter  of  the  cord  which  bears 


Stiffness  of  Cords.  157 

the  weight  being  8,  and  the  radius  of  the  surface  R.  We  wish 
to  know  what  this  addition  must  be,  when  the  weight  is  p',  the 
diameter  of  the  cord  d',  and  the  radius  of  the  surface  R'.  It  will 
be  observed,  after  what  has  been  said,  that  if  there  were  no  dif- 
ference except  with  respect  to  the  entire  weight  by  which  the 
cord  is  stretched,  we  should  arrive  at  a  solution  by  the  propor- 
tion 

p  :  p'  :  :  k  :  —-  =  the  addition  required. 

But  if,  beside  the  difference  in  the  weights,  there  is  also  a  differ- 
ence in  the  curvature  of  the  surfaces ;  then,  by  the  second  of  the 
above  remarks  ;  namely,  that  the  additions  arising  from  this  cause 
are  in  the  inverse  ratio  of  the  radii  of  the  surfaces,  we  should  ob- 
tain the  addition  in  question,  together  with  that  due  to  a  change  of 
weight,  by  the  following  proportion, 

z?/  .   7?  .  .  ^  P'    .   ^^P' 
p  K'  p 

Regard  being  had  to  the  third  remark,  we  shall  obtain  the  addi- 
tion to  be  made  on  account  of  the  three  causes  united,  by  the 
proportion 

,      ^,        JcRp'  d'  Jc  R  p' 

8:8'::  -^r^    :  x  =     ,  „,  ^  ■ 
R'  p  8  R'  p 

The  resistance  in  the  first  case,  therefore,  will  be  to  the  resistance 

m  the  second 

J       8'  kRp'  ^  „,  .,  r>    ,  8p      8'p' 

that  is,  the  resistances  arising  from  the  stiffiiess  of  the  cords  are 
as  the  weights  which  stretch  the  cords,  multiplied  by  the  diame- 
ters of  these  cords,  and  divided  by  the  radii  of  the  surfaces  over 
which  they  pass. 

These  conclusions,  it  may  be  observed,  are  not  perfectly  rig- 
orous ;  but  they  may  be  regarded  as  sufficiently  exact  for  prac- 
tice, till  experiment  has  thrown  new  light  upon  the  subject.  In- 
deed, experiment  shows  that  the  resistance  arising  from  the  stiffness 


158  Statics. 

of  cords,  agrees  nearly  with  this  law ;  but  all  the  experiments  that 
have  been  made  upon  this  subject  have  not  hitherto  agreed  so  per- 
fectly with  the  theory  as  might  be  wished. 

255.  We  shall  now  illustrate  the  foregoing  principles  by  an 
example.  For  this  purpose,  let  us  suppose  the  common  r^adius 
Fig.  92.  of  the  pulley  to  be  2  inches,  that  of  the  axle  I  of  the  same  quanti- 
ty, and  the  diameter  of  tl)e  cords  to  be  ~  of  an  inch.  Let  us 
take,  moreover,  the  result  of  experiments  on  this  subject,  namely, 
that  a  cord  of  half  an  inch  diameter,  loaded  with  120"'-,  and  pass- 
ing over  a  pulley  of  |  of  an  inch,  occasions,  by  its  stiffness,  a  re- 
sistance of  b"*-. 

This  being  established,  the  weight  p  being  400"'-,  as  in  article 
242,  the  branches  1  and  2  will  be  loaded,  both  on  account  of 
this  weight  and  the  force  added  to  q  to  overcome  the  friction,  by 
a  force  equal  to  200"'-,  G.  Multiplying,  therefore,  this  weight  by 
^  of  an  inch,  the  diameter  of  the  cord,  and  dividing  by  2  inches, 
the  radius  common  to  all  the  pulleys ;  multiplying  also  the  weight 
120"'-,  used  in  our  experiment,  by  |,  the  diameter  of  the  cord, 
and  dividing  by  |,  the  radius  of  the  pulley,  in  the  experiment  in 
question  ;  we  shail  have  for  the  results,  34,  93,  and  40.  Now 
in  order  to  obtain  the  force  required  by  the  stiffness  of  the  cord 
1,  2,  which  passes  beneath  the  movable  pulley,  we  must  use  this 
proportion, 

40  :  34,  93  :  :  S"'-  :  6,  99  or  T"*-  nearly  ; 

this  fourth  term  is  the  quantity  which  it  would  be  necessary  to 
add  to  the  tension  of  the  cord  2,  if  the  lower  pulley  were  fixed  ; 
but  since  it  is  movable,  which  diminishes  the  tension  by  |,  as 
we  shall  see  below,  we  have  only  3"^-,  5  to  be  added  to  109"'-,  6, 
by  which  this  cord  is  already  stretched  ;  the  whole  tension  will 
thus  be  lis"'-,  1. 

242.  We  have  seen  that,  on  account  of  friction,  the  tension  of  the 

cord  3  ought  to  be  equal  to  120"^-,  7  ;  therefore  the  whole  force 
by  which  the  cord  2,  3,  which  passes  over  the  fixed  block,  is  ex- 
tended, is  113,  1  -f-  120,  7  on  233"'-,  8.  Multiplying  this  force  as 
above,  by  i,  and  dividing  by  2,  we  shall  have  38,  97.  Seeking  as 
before  the  value  of  the  stiffness  of  the  cord  2,  3,  which  passes  over 
the  fixed  block,  we  shall  have  the  proportion, 


Stiffness  of  Cords.  1 59 

40  :  38,  97  :  :  &>-  :  7«'-,  79. 

Adding  now  these  7"'-,  79  to  120"'-,  7,  the  quantity  before  found 
for  the  tension  of  the  cord  3,  we  shall  have  12S"'-,  5  for  the  force 
of  tension,  both  on  account  of  friction  and  the  stiffness  of  the 
cords. 

The  cord  4,  on  account  of  friction  is  stretched  by  a  force  '^^^' 
=  130"'-,  3;  the  whole  force  of  tension  upon  the  cord  3,  4,  which 
embraces  the  movable  block,  is  therefore  25S"'-,  S  ;  with  this 
quantity,  the  dimensions  of  the  pulleys  and  cords  remaining  the 
same,  we  shall  find,  as  above,  that  the  allowance  for  the  stiffness 
of  the  cords,  would  be  S""-,  62,  if  this  cord  did  not  embrace  the 
movable  block  ;  but  proceeding  as  we  have  done  above,  and  for 
the  same  reason,  we  must  take  but  half  of  this  quantity  j  thus  the 
tension  of  the  cord  4,  all  things  considered,  will  be  134"'-,  6 
nearly. 

The  cord  5,  on  account  of  friction,  is  stretched  with  a  force 
=  143'^-,  5;  the  whole  force  by  which  the  cord  4,  5,  embracing 
the  fixed  pulley,  is  stretched,  is  therefore  27S"'-,  1.  With  this  value, 
the  dimensions  remaining  the  sauie,  we  shall  find  the  force  required 
for  the  stiffness  of  the  cords  to  be  9"'-,  3  ;  the  whole  tension  of  the 
cord,  therefore,  is  152"'-,  8.  Thus  friction  and  the  stiffness  of 
the  cords  together  require  the  weight  g,  which  would  otherwise 
be  but  lOO"*-,  to  be  J53"'-  nearly,  a  quantity  greater  by  more  than 
one  half. 

We  have  taken  only  half  of  what  the  calculation  furnished  as  the 
quantity  to  be  added  to  cords  2  and  4,  on  account  of  the  stiffness. 
The  reason  of  this  is,  that  the  cord  by  which  the  movable  pulley 
is  made  to  revolve,  may  be  considered  as  turning  it  on  a  centre 
placed  at  the  point  where  this  pulley  touches  the  other  cord  ;  and 
consequently  the  case  is  similar  to  that  of  a  fixed  pulley,  having  a 
radius  equal  to  the  diameter  of  the  movable  pulley  ;  and  since  the 
forces  required  on  account  of  the  stiffness  of  the  cords,  are  in  the 
inverse  ratio  of  the  radii  of  the  pulleys,  the  force  in  this  case  must 
be  diminished  one  half. 


i  60  Statics. 


Of  the  Balance,  Steelyard,  ^c. 

Fig.  142.  256.  A  balance  is  a  lever  of  the  first  kind  in  which  the  fulcrum 
or  axis  is  in  the  middle  between  the  points  of  application  of  the 
forces  or  tveights.  It  is  apparent  that  with  such  an  instrument,  a 
state  of  equilibrium  must  indicate  an  equality  in  the  weights  lo  be 
compared. 

There  are  several  particulars  to  be  attended  to  in  the  con- 
struction of  a  good  balance.  (1.)  The  axis  should  be  above  and 
not  too  far  above  the  centre  of  gravity  of  the  two  arms  or  beam  ; 
for  if  it  pass  through  this  centre  of  gravity,  the  beam,  when  loaded 
with  equal  weights,  or  not  loaded  at  all,  will  remain  at  rest  in 
any  position  whatever ;  and  it  is  intended  that  a  horizontal  position 
shall  indicate  equal  weights.  If  the  axis  be  below  the  centre  of 
gravity  of  the  beam,  the  slightest  deviation  of  this  centre  from  a 
vertical  position  over  the  axis  would  be  followed  with  a  motion  that 
would  tend  to  reverse  the  position  of  the  beam.  Moreover,  if  the 
axis  be  at  a  considerable  distance  above  the  centre  of  gravity,  the 
beam  would  have  too  great  a  tendency  to  a  horizontal  position, 
independently  of  an  equality  in  the  weights,  and  would  according- 
ly want  the  requisite  sensibility. 

(2.)  The  axis  and  points  of  a[)plication  of  the  weights,  or  points 
of  suspension,  should  be  in  the  same  straight  line  ;  otherwise  the 
beam  when  loaded  is  liable  to  the  defects  just  mentioned  by 
having  its  centre  of  gravity  sliifted  above  the  axis  or  too  far  below 
it.  It  is  accordingly  of  great  importance  that  the  arms  should 
be  so  constructed  as  not  to  bend  or  yield  in  any  degree  in  con- 
sequence of  the  weights  attached  to  them,  while  at  the  same 
time  they  should  be  as  light  as  possible.  To  secure  both  these 
objects,  in  the  best  balances  instrument-makers  have  given  to 
the  arms  the  form  of  hollow  cones.  Care  is  taken  also  to  pre- 
serve the  equality  in  point  of  length  of  the  two  arms,  and  to  di- 
minish friction,  by  making  the  axis  and  points  of  suspension  of 
hardened  steel,  and  in  the  shape  of  a  very  acute  wedge,  or  knife- 


Balance,  Steelyard,  <^c.  161 

edge,  the  plane  surfaces  with  which  they  come  in  contact  being 
likewise  of  the  hardest  substances. 

257.  Nevertheless,  where  extreme  accuracy  is  required,  the 
best  method  is  to  place  the  thing  to  be  weighed  in  one  of  the 
scales,  and  to  balance  it  by  any  convenient  substances  in  the  other 
scale,  till  the  beam  takes  an  exact  horizontal  position,  as  shown  by 
the  index  FL*  Then  carefully  remove  the  thing  whose  weight 
is  sought,  and  replace  it  by  accurate  weights,  as  grains,  he,  till 
the  beam  takes  precisely  the  same  position.  The  number  of 
grains,  &c.,  used  will  be  the  weight  required.  In  this  way,  all 
error  arising  from  the  want  of  perfect  equality  in  the  two  arms  of 
the  balance  is  completely  obviated. 

258.  With  all  these  precautions,  the  process  of  weighing  is  but 
an  approximation,  strictly  speaking,  to  perfect  accuracy.  (1.)  The 
best  balance  tends  of  itself  with  some  degree  of  force  to  a  horizon- 
tal position,  since  the  slightest  inclination  must  cause  the  centre 
of  gravity  to  rise  somewhat,  and  consequently,  if  the  difference  in 
the  two  weights  does  not  exceed  this  force,  the  balance  will  not 
enable  us  to  detect  the  inequality.  (2.)  Tiie  friction,  both  of  the 
axis  and  of  the  points  of  suspension,  is  an  obstacle  to  motion,  and 
the  difference  in  the  weights  must  be  sufficient  to  overcome  this, 
in  addition  to  the  force  just  mentioned,  in  order  that  the  balance 
may  show  them  to  be  unequal. 

Knowing,  however,  what  weight  is  sufficient  to  produce  mo- 
tion in  any  case,  we  know  what  degree  of  accuracy  we  have 
been  able  to  attain.  This  is  different  for  different  weights.  A 
small  weight  can  be  determined  more  exactly,  other  things  being 
the  same,  than  a  large  one,  since  the  friction  increases  with  the 
pressure.  A  balance  constructed  by  Ramsden  for  the  Royal  Soci- 
ety of  London,  when  loaded  with  a  weight  of  lO"'-,  would  turn 
with  a  ten-millionth  part  of  the  weight,  or  a  little  more  than  a  200lh 
part  of  a  grain  ;  and  one  made  by  Fortin  of  Paris,  when  charged 
with  a  weight  of  4'^-,  would  turn  with  a  70th  part  of  a  grain. 

*  Instead    of  an  index  rising  perpendicularly,   a   pin   is  some- 
times attached  to  the  end  of  the  beam,  and  made  to    move   over   a 
nicely  graduated  arc  provided  with  a   microscope  ;  and    the    whole 
apparatus  enclosed  under  glass  to  prevent  any  agitation  from  the  air. 
Mech.  2  J 


1 62  Statics. 

Fig,  143.  259.  The  steelyard  also  is  a  lever  of  the  first  kind;  but  the 
fulcrum  or  axis  of  this  instrument  divides,  the  distance  between 
the  points  of  application  of  the  weights  into  unequal  parts  ;  and 
instead  of  different  weights  placed  at  the  same  distance,  a  con- 
stant weight  or  poize  is  placed  at  different  distances  to  effect  an 
equilibrium  with  the  article  to  be  weighed  ;  and  the  weight  of  this 
article  will  be  to  that  of  the  poize  inversely  as  their  distances  from 
the  axis.  When  these  distances  are  equal,  the  weight  of  the  poize 
is  equal  to  that  of  the  article  weighed.  At  double  ibis  distance, 
the  poize  indicates  a  weiiiht  double  its  own,  and  at  half  this  distance 
the  ueiglit  indicated  is  half  that  of  the  poize,  and  so  on.  Tlie  lon- 
ger arm  of  the  steelyard,  therefore,  being  graduated  upon  this  prin- 
ciple, it  becomes  a  convenient  and  sufficiently  accurate  instrument 
for  weighing  all  gross  substances. 

The  same  rule  is  to  be  observed  with  respect  to  the  position 
of  the  axis  of  motion  in  the  construction  of  the  steelyard,  as  in  that 
of  the  balance  ;  that  is,  the  axis  must  be  in  a  line  with  the  points 
of  suspension,  and  a  little  above  the  centre  of  gravity,  so  that  the 
arms  when  unloaded,  or  when  equal  moments  are  indicated,  shall 
preserve  a  horizontal  position. 

Fig.144,  2G0.  In  the  bent-lever  balance  represented  in  figure  144,  instead 
of  a  movable  weight  resting  upon  a  horizontal  beam,  the  natural 
weight  of  an  inclined  arm  is  made  use  of,  and  is  drawn  out  to 
different  distances  from  a  vertical  position  according  to  the  weight 
attached  to  the  other  arm.  By  applying  different  known  weights 
to  the  arm  A,  and  graduating  the  arc  CD  according  to  the  posi- 
tions of  the  arm  B,  we  shall  have  an  instrument  very  analogous 
to  the  steelyard  ;  for  the  weight  JB  may  be  considered  as  shifted 
to  different  distances  alodg  the  horizontal  line  FD,  since  it  would 
have  precisely  the  same  effect  here  as  in  its  actual  position.  Con- 
sequently, if  the  direction  of  the  weight  attached  to  the  arm  A 
preserved  always  the  same  horizontal  distance  from  the  axis  of 
motion  (which  it  might  be  made  to  do,  by  suspending  it  from  a  sin- 
gle cord  applied  to  the  arc  of  a  circle  having  the  axis  for  its  centre,) 
the  arc  CZ>  might  be  graduated  by  dividing  the  radius  DF  into 
equal  parts  in  the  manner  of  the  steelyard,  and  then  transferring 
these  divisions  with  the  numbers  denoting  the  weights  to  the  arc 
CD,  by  means  of  vertical  lines. 


Balance,  Steelyard,  ^-c.  1G3 

261.  Sometimes  the  axis  is  made  to  cany  a  wheel,  the  teeth  of 
which  act  upon  a  pinion  furnished  with  an  index  and  dial-plate, 
whereby  a  slight  motion  is  rendered  very  conspicuous. 

The  bent-lever  balance  is  better  adapted  to  despatch  than  any 
other  instrument  for  weighing.  But  it  is  liable  to  irregularity,  and 
is  not  susceptible  of  the  accuracy  of  the  balance. 


DYNAMICS. 


Of  Motion  uniformly  accelerated. 

262.  A  body,  perfectly  free,  having  once  received  an  itnpulse 
will  continue  its  motion,  the  velocity  and  the  direction  remaining 
always  the  same  as  at  the  first  instant.  But  if  it  receive  a  new 
impulse  in  the  same  direction,  or  in  a  direction  opposfte  to  the 
first,  it  will  move  with  a  velocity  equal  in  the  former  case  to  the 
sum,  and  in  the  latter  to  the  difference,  of  the  velocities  it  has 
successively  received. 

Hence,  if  we  suppose  that,  at  determinate  intervals  of  time, 
the  body  receives  new  impulses  in  the  same  direction,  or  in  a 
direction  opposite  to  the  first,  it  will  have  a  varied  or  unequal 
motion,  and  its  velocity  will  change  or  become  different  at  the 
commencement  of  each  interval  of  time. 

Whatever  this  may  be,  the  velocity  of  a  body  at  the  end  of 
any  given  period  of  time,  is  to  be  estimated  by  the  space  which 
this  body  is  capable  of  describing  in  a  unit  of  time  on  the  supposition 
that  its  motion  becomes  uniform  at  the  instant  from  which  this 
velocity  is  to  be  reckoned. 

Any  force,  which  acting  upon  a  body  causes  it  to  vary  its 
motion,  is  called  an  accelerating  or  retarding  force.  When  it  acts 
equally  at  equal  intervals  of  time,  it  is  called  a  uniformly  accelerating, 
or  uniformly  retarding  force,  according  as  it  tends  to  increase  or 
diminish  the  actual  velocity  of  the  body. 

We  shall  now  examine  the  circumstances  of  motion  uniformly 
accelerated. 

263.  Since  in  this  kind  of  motion,  the  accelerating  force  acts 
always   in  the   same  manner,  if  we   suppose  that  u  is  the  velocity 


17. 


1 66  Dynamics. 

communicated  at  each  unit  of  time,  it  is  evident  that  the  successive 
velocities  of  the  hody  will  be  u,  2  v,  3  u,  &c.,  so  that  after  a 
number  of  units  of  time,  denoted  by  i,  the  velocity  acquired  will 
be  n,  taken  as  many  times  as  there  are  units  in  t  ;  that  is.  it  will  be 
t  X  It    or  t  u. 

264.  Hence,  (1.)  In  the  case  of  niotion  uniformly  accelerated, 
the  number  of  degrees  of  velocity  which  the  body  acquires,  increases 
as  the  number  of  intervals  or  periods  during  which  the  motion 
continues,  which  may  be  expressed  by  saying  that  for  different 
times  the  velocities  acquired  are  as  the  times  elapsed  from  the 
commencement  of  the  motion. 

Thus,  if  we  call  v  the  velocity  that  the  body  acquires  at  the 
end  of  the  time  t,  we  shall  have 

V  =  u  t. 

(2.)  The  velocities  which  the  body  will  be  found  to  have 
during  the  lapse  of  each  successive  interval,  will  form  an  arithmeti- 
cal progression  or  progression  by  differences, 

-r-  u   .   2  u   .    3  u   .   &ic. 

the  last  term  of  w-hich  is  m  i  or  v,  and  of  which  the  number  of 
terms  is  t,  that  is,  is  denoted  by  the  number  of  actions  of  the  accele- 
rating force. 

(3.)  Also,  since  the  velocities  u,  2  «,  he,  are  simply  the  spa- 
22.  ces  that  the  body  would  describe  in  the  corresponding  intervals 
of  time  respectively ;  the  whole  space  described  during  the  time 
t  will  be  the  sum  of  the  terms  u  -{-  2  u  -\-  &;c.,  of  this  progression, 
Aig.229.  that  is,  it  will  be  expressed  by  (u  -f-  ^)  X  |  <•  Therefore,  if  we 
call  s  this  whole  space  passed  over  from  the  commencement  of  the 
motion,  we  shall  have 

s  =  (u+v)  X  ^t. 

265.  Let  us  suppose  now,  that  the  accelerating  force  acts 
without  interruption,  or,  which  amounts  to  the  same  thing,  that 
the  time  is  divided  into  an  infinite  number  of  infinitely  small  parts, 
which  we  shall  call  instants ;  and  that  at  the  beginning  of  each 
instant,  the  accelerating  force  exerts  a  new  impulse  upon  the 
body.     Let  us  suppose,  also,  that  it  acts  by  degrees  infinitely  small. 


Accelerated  Motion.  167 

Then,  u  being  infinitely  small  with  respect  to  v,  which  is  the 
velocity  acquired  during  the  infinite  number  o(  instants  denoted 
by  t,  u  is  to  be  neglected  in  the  equation  s  =■  {u  -^  v)  X  ^  t 
whicli  gives  ^^^-  ^• 

s  =  ^  V  t. 

266.  This  being  established,  let   us  suppose   that  at  the   end 

of  the  time  t,   the  accelerating  force    ceases  to  act ;  tlie  body   will     17. 
continue  its  motion  w'ith  tlie  velocity  v  that  it  had   acquired  ;  that  is, 
in  each  unit  of  time,  it  will   describe  a  space  equal  to  v  ;  according-     22. 
]y,    if    it    were    to    continue    with    this    velocity    during    the    time 
t,   it  would  describe  a  space   equal  to   i?    X    ^  that  is,   double  the 
space  s  or  ^  V  t,  described  in  the  same  time  by  the  successive  action     265. 
of  the    accelerating   force.     Therefore,    in  motion   uniformly   and 
continually  accelerated.,  the  space  described  during  a    certain  lime  is 
half  of  that  ivhich  the   body  uould  describe  in    an    equal    time    with 
the  last  acquired  velocity  continued  uniformly. 

267.  Since  the  acquired  velocity  increases  with  the  limes 
elapsed,  if  we  call  g  the  velocity  acquired  at  the   end  of  a  second, 

the  velocity  acquired  after  a  number  t  of  seconds,  will  be  g  t;  that     264. 
is,  we  have, 

V  =  g  t; 

and  accordingly  the  equation  s  =z  i  v  t,  found  above,  becomes 

s  =  ^gt  X  t  =  -lgt\ 

If,  therefore,  we  represent  by  5'  another  space  described  in  the 
same  manner  during  a  time  t',  we  shall  have,  according  to  the 
above  reasoning, 

S'  =:lgt'^, 

from  which  we  deduce  the  proportion, 

s  :  s'  ::  Igt^    :  ^  g  t'^    :  :  t^    :  t'^  ; 

we  hence  learn  that,  with  respect  to  motion  uniformly  accelerated, 
the  spaces  described  are  as  the  squares  of  the  times. 

26S.    Moreover,   since  the  velocities  are  as  the  limes,   we   con-     264. 
elude   also,    that   the   spaces    described    are  as  the  squares  of  the 
velocities. 


168  Dynamics. 

269.  Therefore  the  velocities  and  the  times  are  each  as  the 
sqvare  roots  of  the  spaces  described  from  the  commencement  of  the 
motion. 

270.  The  principles  here  established  are  equally  applicable  to 
the  case  of  moiioii  uniformly  retarded,  provided  that  by  llie  times 
we  understand  those  which  are  to  elapse,  and  by  the  spaces  those 
which  are  to  be  described,  from  the  instant  in  question  till  the 
velocity  is  destroyed. 

271.  From  the  equation  s  =  -k  S  ^^^  ^^i^  quantity  g,  by  which 
we  have  understood  the  velocity  that  the  accelerating  force  is  capa- 
ble of  pi'oducing  by  its  action,  exerted  successively  during  a  second 
of  lime,  is  what  we  call  the  accelerating  force,  since  we  must  judge  of 
this  force  by  the  effect  which  it  is  capable  of  producing  in  a  body  in 
a  determinate  time,  an  effect  which  is  nothing  else  but  the  communi- 

27.     cation  of  a  certain  velocity. 

Of  free  Motion  in  heavy  Bodies. 

272.  It  is  the  kind  of  motion  we  have  been  considering,  to 
which  the  motion  of  heavy  bodies  is  to  be  referred.  But  before 
applying  to  this  subject  the  theory  above  developed,  it  will  be 
proper  to  make  known  a  few   facts  concerning  gravity,  in   addition 

'''4.     to  those  heretofore  given. 

As  to  the  magnitude  of  the  force  of  gravity,  it  is  different, 
strictly  speaking,  in  different  latitudes  and  at  different  distances  from 
the  centre  of  the  earth  in  the  same  latitude.  But  the  quantities 
by  which  it  varies,  as  we  depart  from  the  equator,  are  very  small, 
and  do  not  in  any  manner  concern  us  at  present.  The  same  may 
be  said  of  the  variations  it  undergoes,  according  as  we  rise  above, 
or  descend  below,  the  mean  surface  of  the  earth  ;  they  cannot 
become  sensible,  except  by  changes  of  distance  much  more  con- 
siderable than  any  to  which  we  are  accustomed  ;  so  that  for  the 
present  we  may  regard  gravity  as  a  force  every  where  the  same, 
or  one  which  urges  bodies  downward  by  the  same  quantity  in  the 
same  time. 

This  force  is  to  be  considered  also  as  acting,  and  acting 
equally  at  each  instant,  upon  every   particle   of  the   matter  about 


Motion  of  heavy  Bodies.  169 

us.  Now  it  is  evident,  that  if  each  of  the  parts  of  a  body  receive 
the  same  velocity,  the  whole  will  move  only  with  the  velocity 
that  any  detached  portion  would  have  received ;  so  that  the 
velocity  which  gravity  impresses  upon  any  mass  whatever,  does 
not  depend  upon  the  magnitude  of  this  mass  ;  it  is  the  same  for 
a  large  body  as  for  a  small  one.  It  is  true,  however,  that  all 
bodies  are  not  observed  to  fall  from  the  same  height  in  the  same 
time  ;  but  this  difference  is  the  effect  of  the  resistance  of  the  air, 
as  we  shall  see  hereafter ;  and  when  bodies  are  made  to  fall  in 
close  vessels,  from  which  the  air  has  been  withdrawn,  though  of 
very  different  masses,  they  are  found  to  descend  through  the  same 
space  in  the  same  time. 

It  may  be  well  to  notice  here  the  distinction  between  the  effect 
of  gravity  and  that  of  weight.  The  effect  of  gravity  is  to  cause,  or 
tend  to  cause  in  each  part  of  matter,  a  certain  velocity,  which  is 
absolutely  independent  of  the  number  of  material  particles.  But 
weight  is  equal  to  the  effort  necessary  to  be  exerted,  in  order  to 
prevent  a  given  mass  from  obeying  its  gravity.  Now  this  effort 
depends  upon  two  things ;  namely,  the  velocity  that  gravity  tends  to 
cause  in  each  part,  and  the  number  of  parts  on  which  this  force 
is  exerted.  But  as  the  velocity  which  gravity  tends  to  give,  is  the 
same  for  each  part  of  matter,  the  effort  to  be  exerted  in  order  to 
prevent  a  given  mass  from  obeying  its  gravity,  is  proportional  to  the 
number  of  parts,  that  is,  to  its  mass.  Thus  weight  depends  upon 
the  mass,  whereas  gravity  has  no  relation  to  it. 

273.  Having  made  known  these  particulars  with  regard  to 
gravity,  we  proceed  to  the  laws  of  motion  of  heavy  or  gravitating 
bodies. 

Since  gravity  acts  equally  and  without  interruption,  at  what- 
ever distance  the  body  is  from  the  surface  of  the  earth  (at  least, 
so  far  as  our  experience  extends),  gravity  is  a  uniformly  accele- 
rating force,  which  at  each  instant  causes  in  a  body  a  new  degree 
of  velocity,  that  is  always  the  same  for  each  equal  instant;  so 
that  the  velocities  acquired,  increase  as  the  times  elapsed ;  the 
spaces  passed  over  are  as  the  squares  of  the  times,  or  as  the 
squares  of  the  velocities  ;  the  velocities  are  as  the  square  roots 
of  the  spaces  described ;  the  times  are  also  as  the  square  roots 
of  the  spaces  described  ;  in  short,  all  that  we  have  said  respect- 
Mech.  22 


170  Dynamics. 

ing  a  uniformly  accelerating  force,  is  strictly  applicable  to  gravity, 
it  being  well  understood  at  the  same  time,  that  the  resistance  of 
the  air  and  obstructions  of  every  kind  are  out  of  the  question. 

In  order  to  determine,  therefore,  with  respect  to  the  motion 
of  heavy  bodies,  the  spaces  described  and  the  velocities  acquired, 
we  require  only  one  single  effect  of  gravity  for  a  determinate 
time.  For  the  equations  v  =z  g  t,  s  =i  ^  g  t^,  enable  us  to  calcu- 
late each  of  the  particulars  above  enumerated,  when  the  value  of  g 
is  known. 

It  must  be  recollected  that  by  g  we  have  understood  the 
velocity  which  a  body  acquires  by  gravity  in  one  second  of  time. 
267.  Now  we  know  by  actual  observation,  that  a  body,  not  impeded  by 
the  resistance  of  the  air  or  other  obstacle,  falls  at  the  surface  of  the 
earth  through  16,1  feet  in  one  second.  We  shall  see  hereafter  how 
this  is  determined. 

But  we  have  shown  that  with  the  velocity  acquired  by  a 
series  of  accelerations,  the  body  would  describe  with  a  uniform 
266.  motion  double  the  space  in  the  same  time.  Hence  the  veloc- 
ity acquired  by  a  heavy  body  at  the  end  of  the  first  second  of 
its  fall  is  such,  that  if  gravity  ceased  to  act,  it  would  describe 
twice  16,1  feet,  or  32,2  in  each  succeeding  second.  Therefore 
g  =  32,2  feet. 

274.  Now  of  the  two  equations  v  z=z  g  t,  and  s  =■  \  g  t^,  the 
first  teaches  us  that  in  order  to  find  the  velocity  acquired  by  a  heavy 
body,  falling  freely,  during  a  number  t  of  seconds,  it  is  necessary  to 
multiply  the  velocity  acquired  at  the  end  of  the  first  second  by  the 
time  t,  or  number  of  seconds. 

Hence,  when  a  heavy  body  has  fallen  during  a  certain  number 
of  seconds,  the  velocity  acquired  is  such,  that  if  gravity  ceased  to  act, 
the  body  woidd  describe  in  each  second  as  many  times  32,2  feet,  as 
there  were  seconds  elapsed. 

Thus  a  body  that  has  fallen  during  7  seconds,  will  move  at  the 
end  of  the  7  seconds,  with  a  velocity  equal  to  7  times  32,2  or 
225,4  feet  in  a  second  without  any  new  acceleration. 

275.  From  the  second  of  the  above  equations,  namely, 


Motion  of  heavy  Bodies.  171 

we  learn  that  in  order  to  find  the  space  or  height  s  through  which 
a  heavy  body  falls  in  a  number  t  of  seconds,  we  have  only  to 
multiply  the  square  of  this  number  of  seconds  by  i  g,  that  is,  by 
the  space  described  in  the  first  second. 

Hence,  the  height  or  number  of  feet  through  which  a  heavy  body 
falls  during  a  number  t  of  seconds  is  so  many  times  16,1  feet  as 
there  are  units  in  the  square  of  this  number  of  seconds. 

Thus,  when  a  body  has  been  suffered  to  fall  freely  during  7 
seconds,  we. may  be  assured  that  it  has  passed  through  a  space 
equal  to  49  times  16,1  feet,  or  788,9  feet.  We  see,  therefore, 
that  when,  in  the  case  of  falling  bodies,  the  time  elapsed  is  known, 
nothing  is  more  easy  than  to  determine  the  velocity  acquired,  and 
the  space  described. 

276.  If  the  question  were  to  find  the  time  employed  by  a  body 
in  falling  from  a  known  height,  the  equation  s  =z  ^  g  t^, 

gives  t^  :=  -^ — ,  and  consequently, 
2-  S 


=Jf 


t  ■   ' 


that  is,  we  seek  how  many  times  the  height  s  contains  ^  g,  or  16,1 
feet,  the  space  described  by  a  body  in  the  first  second  of  its  fall, 
and  take  the  square  root  of  this  number. 

277.  If  we  would  know  from  what  height  a  heavy  body  must  fall 
to  acquire  a  given  velocity,  that  is,  a  velocity  by  which  a  certain 
number  of   feet  is  uniformly  described  in  a  second ;   from  the 

equation  v  =  gt,l  deduce  the  value  of  t,  namely,  ^  =  -  5  substi- 
tuting this  value  in  the  equation  s  =  \  g  t^,  1  have 

v^         v^ 

by  which  I  learn,  that  in  order  to  find  the  height  s  from  which  a 
heavy  body  must  fall  to  acquire  a  velocity  v,  of  a  certain  number 
of  feet  in  a  second,  the  square  of  this  number  of  feet  is  to  be 
divided  by  double  the  velocity  acquired  by  a  heavy  body  in  one 
second,  that  is,  by  64,4. 


172  Dynamics. 

Thus,  if  I  would  know,  for  example,  from  what  height  a  heavy 
body  must  fall,  to  acquire  a  velocity  of  100  feet  in  a  second,  I 
divide  the  square  of  100,  namely,  10000,  by  G4,4  ;  and  the  quo- 
tient VV,!"  =^  155,2  he,  is  the  height  through  which  a  body  must 
fall  to  acquire  a  velocity  of  100  feet  in  a  second. 

We  might  evidently  make  use  of  the  same  formula  in  deter- 
mining to  what  height  a  body  would  rise,  when  projected  vertically 
upward  with  a  known  velocity. 

Moreover,  from  the  above  equation,  s  =  ^r—  we  obtain 

^ 2^ 

v^  :=  2  g  s,  or  V  =.  \/2  g  s  =  8,024  \/s, 

that  is,  the  velocity  acquired  in  falling  through  any  space  s,  is 
equal  to  \/2g  s,  or  equal  to  eight  times  the  square  root  of  s  nearly, 
V,  g,  and  5,  being  estimated  in  feet.  Thus  the  velocity  acquired  in 
falling  through  1  mile  or  5280  feet,  is  equal  to 

8,024  V5280  =  583  feet  very  nearly. 

278.  By  these  examples  it  will  be  seen  that  all  the  circum- 
stances of  the  motion  of  heavy  bodies  may  be  easily  determined; 
and  it  is  accordingly  to  these  motions,  that  we  commonly  refer 
all  others;  so  that,  instead  of  giving  immediately  the  velocity  of 
a  body,  we  often  give  the  height  from  which  it  must  fall  to  ac- 
quire this  velocity.  Occasions  will  be  furnished  for  examples 
hereafter. 

We  will  merely  observe,  therefore,  by  way  of  recapitulation, 
that  all  the  circumstances  of  accelerated  motion,  and  consequent- 
ly of  the  motion  of  heavy  bodies,  are  comprehended  in  the  two 
equations  v  =  g  t,  s  z=  \  g  f ;  so  that,  g  being  known,  and  one 
of  the  three  things,  t,  s,  v,  or  the  time,  space,  and  velocity,  the  two 
others  may  always  be  found,  either  immediately  by  one  or  the  other 
of  the  above  equations,  or  by  means  of  both  combined  after  the 
manner  of  article  277. 

279.  When  a  body  is  subjected  to  the  action  of  a  force  that 
is  exerted  upon  it  without  interruption,  but  in  a  different  manner 
at  each  successive  instant,  we  give  to  the  motion  the  general  de- 
nomination of  varied.  We  have  examples  of  varied  motion  in 
the  unbending  of  springs ;  although  in  this  case  the  velocity  goes 


Motion  of  heavy  Bodies.  173 

on  increasing,  still  the  degrees  by  which  it  increases  go  on  dimin- 
ishing. The  same  may  be  observed"  with  respect  to  the  degrees  by 
which  the  motion  of  a  ship  arrives  at  uniformity ;  the  action  of  the 
wind  upon  the  sails  diminishes  according  as  the  ship  acquires 
motion,  because  it  is  withdrawn  so  much  the  more  from  this  action, 
according  as  it  has  more  velocity. 

280.  The  principles  necessary  for  determining  the  circum- 
stances of  this  kind  of  motion  are  easily  deduced  from  the  princi- 
ples that  we  have  laid  down  with  regard  to  uniform  motion,  and 
motion  uniformly  accelerated. 

In  whatever  manner  motion  is  varied;  if  we  consider  it  with 
respect  to  instants  infinitely  small,  we  may  suppose  that  the  ve- 
locity does  not  change  during  the  lapse  of  one  of  these  instants. 
Now,  when  the  motion  is  uniform,  the  velocity  has  for  its  expres- 
sion the  space  described  during  any  time  t,  divided  by  this  time. 
Accordingly,  when  the  motion  is  uniform  only  for  an  instant,  the 
velocity  must  have  for  its  expression  the  infinitely  small  space 
described  during  this  instant  divided  by  this  instant.  Hence,  if  s 
represents  the  space  described,  in  the  case  of  a  variable  motion, 
during  any  time  t,  d  s  will  represent  the  space  uniformly  described 
during  the  instant  d  t',  we  have,  therefore, 

V  =  -j—  or  d  s  =.  V  d  t, 
d  t 

as  the  first  fundamental  equation  of  varied  motion. 

281.  In  the  equation  v  =  g  t,  we  have  understood  by  g  the 
velocity  which  the  accelerating  force  is  capable  of  giving  to  a  body 
in  a  determinate  time,  as  one  second,  by  an  action  that  is  supposed 
to  continue  constantly  the  same.     In  the  equation 

d  V  =:  g  d  t, 

the  same  thing  is  to  be  understood.  But  we  must  observe  that 
the  accelerating  force  being  supposed  to  be  variable,  the  quanti- 
ty g  which  represents  the  velocity  that  the  accelerating  force 
is  capable  of  producing,  if  it  were  constant  for  one  second,  this 
quantity  g,  I  say,  is  different  for  all  the  different  instants  of  the 
motion.  Indeed,  it  will  be  readily  conceived,  that  when  the 
accelerating    force    becomes    less,   the  velocity  that  it  is  capable 


1'3'4  Dynamics. 

of  generating  in  a  second  by  its  action  repeated  equally  during  eac 
instant  of  this  second,  must  be 'less,  and  vice  versa. 

282.  From  the  two  equations  d  s  =z  v  d  t,  d  v  =.  g  d  t,WQ  can 

obtain   a   third   that    may   be   employed    with   advantage.      Thus 

d  s 
from  the  equation  d  s  =.  v  d  t,vie  deduce  d  t  =  —  ;  substituting 

this  value  instead  o(  dt  in  the  equation  d  v  =:  g  d  t,  we  have, 

,  d  s 

or, 

g  d  s  z=z  V  d  V. 

283.  We  remark,  that  in  the  process  by  which  we  have  just 
arrived  at  the  equation  d  v  ■=■  g  d  t,  we  regarded  the  velocity  as 
increasing.  If  it  had  gone  on  diminishing,  it  would  have  been 
necessary,  instead  o(  d  v  to  put  —  d  v ;  so  that  the  two  equations 
d  V  :=  g  d  t,  and  g  d  s  z=  v  d  v,io  become  general,  must  be  written 

±  d  V  =:  g  d  t,  ±gdsz=.vdv, 

the  upper  sign  being  used  when  the  motion  is  accelerated,  and  the 
lower  when  the  motion  is  retarded. 

284.  There  is  a  fourth  equation  that  may  be  deduced  from 
the  two  fundamental  equations,  and  which  should  not  be  omitted. 

d  s 
Thus,  the  equation  d  s  =  v  d  t  gives  v  z=  -j—',  whence  we  obtain 


d  t 


^"  =  K^0^ 


substituting  this  value  for  d  v'va  the  equation  g  d  t  ■=.  ±  d  v^vie 
have 


g 


dt 


=  *^(li> 


If  we  suppose,  as  we  are  authorized  to  do,  that  d  t\s  constant, 
we  shall  have, 

g  dt   =1    da   —J —  or  g  d  t^   =    ±   d  d  s. 


Collision.  175 

But  it  must  be  recollected  that,  in  the  equation  g  d  t^  =:  ziz  d  d  s, 
it  is  supposed  that  d  t  is  constant.  When  d  t  is  variable,  we  make 
use  of  the  equation 


,,t  =  ±  <j-^). 


Occasions  will  occur  in  which  these  formulas  will  be  of  great 
use.  But  we  must  not  forget  that  the  quantity  g  which  they 
contain,  represents,  for  each  instant,  the  velocity  which  the  ac- 
celerating force  is  capable  of  giving  to  the  moving  body  in  a  known 
interval  of  time,  as  one  second,  if  during  the  second  it  were  to 
act  with  a  uniformly  accelerating  force  ;  so  that  as  each  quantity  g 
measures,  for  each  instant,  the  effect  of  which  the  accelerating  force 
is  capable,  we  shall  give  it,  for  brevity's  sake,  the  name  of  ac- 
celerating force. 


Of  the  direct  Collision  of  Bodies. 

285.  We  suppose,  in  what  follows,  that  no  account  is  taken  of 
the  gravity  of  bodies,  of  friction,  or  other  resistance. 

We  suppose  also  that  the  bodies,  whose  collision  is  the  subject 
of  consideration,  act  the  one  upon  the  other  according  to  the  same 
straight  line,  passing  through  their  centres  of  gravity,  and  that  this 
straight  line,  is  perpendicular  to  the  plane  touching  their  surfaces  at 
the  point  where  they  meet. 

We  shall  consider  bodies  as  divided  into  two  classes,  denomi- 
nated unelastic  and  elastic ;  the  former  are  supposed  to  be  such 
that  no  force  can  change  their  figure  ;  the  latter  are  regarded  as 
capable  of  having  their  figure  changed,  that  is,  of  being  com- 
pressed, but  as  endued  at  the  same  time  with  a  property  by 
which  this  figure  is  resumed  after  the  compressing  force  is  re- 
moved. 

Although  there  are  not  in  nature  bodies  of  a  sensible  mass, 
that  answer  perfectly  to  each  of  these  descriptions,  yet  it  is  only 
by  proceeding  upon  such  suppositions,  that  we  are  able  to  de- 
termine die  action  of  such  bodies  as  are  actually  presented  to  our 
observation. 


176  Dynamics. 


Of  the  direct  Collision  of  unelastic  Bodies. 

286.  Two  unelastic  bodies  which  meet,  (or  one  of  which  falls 
upon  the  other  at  rest,)  communicate,  or  lose,  a  part  of  their 
motion ;  and  in  whatever  manner  this  takes  place,  we  may  al- 
ways, at  the  instant  of  the  collision,  represent  each  body,  accord- 
ing to  the  principle  of  D'Alembert,  as  urged  with  two  velocities, 
one  of  which  remains   after  the  collision,   while  the  other  is  de- 

133.    stroyed. 

287.  Let  us,  in  the  first  place,  suppose  the  two  bodies  to 
move  in  the  same  direction.  That  which  goes  the  faster,  will 
evidently  lose  a  part  of  its  velocity  by  the  collision,  and  the 
other  will  gain  by  it.  Let  m  be  the  mass  of  the  impinging  body, 
and  u  its  velocity  before  collision  ;  n  the  mass  of  the  impinged 
body  (which  may  be  less  or  greater  than  ni),  and  v  its  velocity 
before  collision.  Let  us  suppose  that  the  velocity  u  changes  to 
u'  by  the  collision  ;  m  will  accordingly  have  lost  m — u'.  I  will 
consider  m  as  having,  at  the  instant  of  collision,  the  velocity  u', 
and  the  velocity  u  —  u'.  If  we  suppose,  in  like  manner,  that  v 
becomes  v'  by  the  collision,  n  will  have  gained  v'  —  v,  I  can 
accordingly  consider  it,  at  the  instant  of  collision,  as  having  the 
velocity  v',  in  the  direction  of  the  actual  motion,  and  the  velocity 
v'  —  v\n  the  opposite  direction  ;  since,  on  this  supposition,  it  will 
really  have  only  the  velocity  v'  —  {y'  —  v)  ox  v. 

As,  therefore,  among  these  four  velocities  there  can,  by  sup- 
position, remain  only  u'  and  v',  the  two  others  u  —  u',  and 
v'  —  V,  must  be  destroyed  in  the  act  of  collision.  Now  as  these 
are  directly  opposite,  it  is  necessary  that  the  quantities  of  motion, 
which  the  bodies  would  have  in  virtue  of  these  velocities,  should  be 
equal ;  we  have,  therefore, 

m  {u  —  u')  z=  n  (v'  —  v). 

Now  in  order  that  u'  and  v'  may,  as  we  have  supposed,  be 
the  velocities  which  the  two  bodies  m  and  n  have  after  collision, 
these  velocities  must  be  such,  that  the  impinging  body  shall  not  have 
the  greater  action  over  the  impinged,  that  is,  that  the  two  bodies 
shall,  after  collision,  proceed  in  company  ;  we  have,  accordingly, 
v'  =.  u'  \  and  hence  form  the  equation. 


33. 


or, 


we  obtain 


Elastic  Bodies.  177 

m  {u  —    w')     =    n  {u'  —  v)i 

mu  —  mu'  =1    nu'    —  n  v, 


m  u  -\-  nv 

u'   =z ; 

m  -\-  n 


therefore,  tvhen  the  bodies  move  in  the  same  direction,  in  order  to 
find  the  velocity  after  collision,  we  take  the  sum  of  the  quantities  of 
motion,  which  the  bodies  had  before  collision,  and  divide  this  sum  by 
the  sum  of  the  masses. 

Thus,  if  m,  for  example,  be  equal  to  5  ounces   and  n   to   7,   u 

equal  to  8  feet  in  a  second,  and  v  to  4  feet  in  a  second,  we  shall 

have, 

_   5X8  +  7X4     _    40  +  28    _,,_.,. 
^    —  5  +  7  ~         12         —    12    —    ^3  J 

that  is,  the  velocity  after  collision  will  be  five  feet  and  two  thirds  in 
a  second. 

288.  If  one  of  the  two  bodies,  as  n  for  example,  were  at  rest 
before  collision,  we  should  have  v  ■=.  0,  and  the  expression  of  the 
velocity  after  collision  would  accordingly  become 


W  = 


m  +  n 


that  is,  we  should  divide    the  quantity  of  motion  belonging  to  the 
impinging  body  by  the  sum  of  the  masses  of  the  two  bodies. 

If,  however,  instead  of  deducing  this  case  from  the  more  gen- 
eral one,  we  would  find  it  directly,  we  should  proceed  according 
to  the  same  principles,  and  consider  the  impinged  body  as  hav- 
ing, in  consequence  of  the  collision,  a  velocity  u',  equal  to  and 
in  the  direction  of  that  which  it  is  to  have  after  collision,  and  a  ve- 
locity —  u',  of  the  same  magnitude,  but  in  the  opposite  direction. 
Thus,  since  it  is  to  preserve  only  the  first,  it  is  necessary  that  in 
virtue  of  the  second  it  should  be  in  equilibrium  with  the  body  m, 
having  a  velocity  u  —  u'  which  it  is  to  lose.  Accordingly  we  must  ^^' 
have 

Mech.  23 


178  Dynamics. 

m  {u  —  u')    =    n  u', 
from  which  we  deduce 


m  u 


m  -\-    n 
the  same  as  the  expression   above  obtained  from  the  general  for- 


mula. 


2S9.  When  the  bodies  move  in  opposite  directions,  in  order  to 
find  the  velocity  after  collision,  it  is  only  necessary  to  suppose  in  the 
first  formula,  that  v  is  negative,  which  gives 


W    — 


m  -\-  n 


that  is,  when  the  bodies  move  in  opposite  directions,  in  order  to  find 
the  velocity  after  collision,  we  take  the  difference  of  the  quantities  of 
motion  belonging  to  the  bodies  before  collision,  and  divide  by  the 
sum  of  the  masses ;  and  this  velocity  will  take  place  in  the  direction 
of  that  body  which  had  the  greater  quantity  of  motion. 

We  might  also  obtain  this  result  directly  by  proceeding  as  in  the 
above  example. 

Thus  the  laws  of  the  direct  collision  of  unelastic  bodies  reduce 
themselves  in  all  cases  to  this  single  rule  ;  the  velocity  after  collision 
is  equal  to  the  sum  or  to  the  difference  of  the  quantities  of  motion 
before  collision  (according  as  the  bodies  move  in  the  same  or  in  op- 
posite directions),  divided  by  the  sum  of  the  masses. 


Of  the  Force  of  Inertia. 

290.  We  have  supposed  in  what  we  have  said,  that  indepen- 
dently of  gravity,  the  resistance  of  the  air,  and  other  obstacles, 
one  of  the  two  bodies  opposes  a  resistance  to  the  other,  and  makes 
it  lose  a  part  of  its  velocity.  But  how  can  a  body  without  gravity, 
and  which  is  confined  by  no  obstacle,  oppose  a  resistance  f 
Does  not  this  seem  to  imply  that  it  would  be  capable  of  giving 
motion  ? 


Force  of  Inertia.  179 

Now  every  resistance  does  not  always  imply  an  actual  motion 
in  the  resisting  body.  If  the  body  A,  for  example,  be  drawn  at 
the  same  time  by  two  equal  and  opposite  forces  represented  by  ./2jB, 
j2C,  it  would  evidently  have  no  motion.  But  it  is  not  less  evident  Fig.i45. 
that  if  a  force  equal  to  CA  were  to  act  upon  it  in  the  direction  C5, 
this  force  would  be  destroyed  by  the  effort  AC,  and  the  body 
would  yield  in  virtue  of  the  force  AB,  equal  to  that  just  applied. 

We  do  not  pretend  to  decide  whether  the  resistance  which 
bodies  oppose  to  motion,  does  or  does  not  arise  from  a  cause  of  this 
kind.  However  the  fact  may  be,  the  resistance  in  question  which 
we  call  the  force  of  inertia,  differs  from  the  resistance  opposed  by 
active  forces  (as  that  of  bodies  which  impinge  against  each  other  in 
opposite  directions)  in  this,  that  these  last  annihilate  a  part  of  the 
motion  ;  whereas,  with  respect  to  the  force  of  inertia,  while  it  de- 
stroys a  part  of  the  motion  in  the  impinging  body,  this  motion 
passes  wholly  into  the  impinged  body,  as  is  clearly  shown  by  the 
equation 

m  {u  —  u')  =■  n  {u'  —  v), 

above  obtained  for  determining  the  motion  after  collision  of  two 
bodies  which  move  in  the  same  direction  ;  for  w  —  u^  is  the  ve-  287. 
locity  lost  by  the  impinging  body,  and  consequently  m  (u  —  u')  is 
the  quantity  of  motion  which  this  body  loses  by  collision.  We 
have,  in  like  manner,  seen,  that  u'  —  v,  is  the  velocity,  and 
n  (u'  —  v)  the  quantity  of  motion,  gained  by  the  impinged  body. 
Now  we  have  shown  that  these  two  quantities  must  necessarily  be 
equal. 

The  force  of  inertia,  therefore,  is,  properly  speaking,  the  means 
of  the  communication  of  motion  from  one  body  to  another.  Every 
body  resists  motion,  and  it  is  by  resisting  that  it  receives  motion ;  it 
receives  also  just  so  much  as  it  destroys  in  the  body  that  acts 
upon  it. 

We  hence  see  that,  every  obstacle  being  removed,  however 
small  we  suppose  the  impinging  body,  and  however  great  the  mass 
impinged,  motion  will  always  take  place  upon  collision.  When,  for 
example,  one  of  the  two  bodies  is  at  rest,  the  velocity  which  has  for 
its  expression 

U'      — i 288. 

m  -\-  n 


180  Dynamics. 

can  never  become  zero,  whatever  be  the  values  assigned  to  m,  n, 
and  u  ;  the  only  case  where  u^  can  be  infinitely  small,  is  that  in 
which  n  is  infinitely  great.  Thus,  if  in  nature  we  see  bodies  lose 
the  motion  that  they  have  received,  it  is  because  they  communicate 
to  the  material  parts  of  the  bodies  which  surround  them.  Now  it 
is  evident  from  the  formula, 


m  -\-  n 

that  the  greater  the  mass  of  the  impinged  body  n,  the  less  (other 
things  being  the  same)  will  be  the  velocity  u',  n  being  considered 
as  the  sum  of  the  material  particles  among  which  m  parts  with  its 
motion.  It  will  be  seen,  therefore,  that  the  velocity  u'  may  soon  be 
so  reduced  as  to  escape  the  notice  of  the  senses,  even  when  it  is 
not  opposed  by  immovable  obstacles,  as  friction,  &ic. 

291.  The  force  of  inertia,  being  a  force  peculiar  to  matter, 
exists  equally  in  every  equal  portion  of  matter,  and  consequently  in 
a  determinate  mass  it  takes  place  according  to  the  quantity  of  mat- 
ter, or  in  proportion  to  the  mass ;  and  as  the  mass  is  proportional  to 
the  weight,  the  force  of  inertia  may  be  regarded  as  proportional  to 
the  weight.  But  we  must  take  care  not  to  infer  hence,  that  the 
force  of  inertia  arises  from  gravity ;  it  is  altogether  independent  of 
it ;  indeed,  if  while  a  body  is  falling  freely,  it  be  forced  forward  by 
the  hand  with  a  velocity  greater  than  that  of  its  natural  descent,  the 
hand  will  experience,  on  overtaking  the  body,  a  blow  or  resistance, 
that  manifestly  cannot  be  attributed  to  gravity,  which  acts  only 
downward.  Still  less  can  it  be  ascribed  to  the  resistance  of  the 
air ;  for  the  resistance  of  the  air,  being  capable  of  acting  only  on  the 
surfaces  of  bodies,  cannot,  like  the  force  of  inertia,  be  proportional 
to  the  quantity  of  matter. 

The  force  of  inertia,  therefore,  is  a  force  peculiar  to  matter,  by 
which  every  body  resists  a  change  of  state,  as  to  motion  and  rest. 
The  force  of  inertia  is  proportional  to  the  quantity  of  matter,  and 
takes  place  in  all  directions  according  to  which  an  effort  is  made  to 
move  a  body. 


Collision  of  unelastic  Bodies.  isi 


Application  of  the  Principles  of  Collision  of  unelastic  Bodies. 

292.  The  principles  which  we  have  laid  down  respecting  the 
collision  of  unelastic  bodies  are  applicable,  whether  the  bodies  im- 
pinge directly  upon  each  other,  as  we  have  supposed,  or  whether 
they  act  upon  each  other  by  means  of  a  rod  which  joins  their 
centres  of  gravity,  or  whether  one  draws  the  other  by  a  thread, 
provided  the  action  is  immediately  and  perfectly  transmitted  to 
the  centre  of  gravity  of  each. 

If,  for  example,  the  two  bodies  m  and  n  act  upon  each  other  Fig.  146. 
by  means  of  a  thread  passing  over  a  pulley  P,  and  we  would  de- 
termine the  motion  that  they  would  receive  in  virtue  of  their  grav- 
ity, we  observe  that  gravity  tends  to  impress  the  same  velocity 
upon  each  of  the  two  bodies  at  each  instant.  Now  as  one  cannot  74 
move  without  drawing  the  other,  the  same  thing  will  take  place 
with  regard  to  the  two  bodies  at  each  new  action  of  gravity,  as  if 
the  two  bodies  drew  each  other  in  opposite  directions  with  equal 
velocities  ;  therefore,  in  order  to  find  the  resulting  velocity,  it  is 
necessary,  calling  u  the  velocity  produced  by  gravity  at  each 
instant  in  a  free  body,  to  take  the  difference  m  it  —  n  u  of  the 
quantities  of  motion,  and  to  divide  it  by  the  sum  m  -\-  n  oi  the 
masses  ;  we  have  accordingly 

m  it  —  n  u  m  —  n 

or      ■ —   u 


m  -f-  n  m  -\-  n 

for  the  actual  velocity  that  each  new  action  of  gravity  would 
give  to  the  body  m.  We  see,  therefore,  since  m,  n,  and  m,  are 
constant  quantities,  that  the  body  m  is  carried  with  a  motion  uni- 
formly accelerated,  and  that  the  force  which  actually  accelerates  it, 
is  to  free  gravity  28. 

tn  —   n  m  —  n         ,  ,      • 

:  :    — — i u    :    u    :  :    ■ —     :    1    :  :    m  —  n    :  m  -{-  n. 

m  -\-   n  m  -\-  n  ' 

Consequently,  if  we  call  g  the  velocity  which  gravity  communi- 
cates to  a  free  body  in  one  second,  we  shall  have  that  which  it 
would  communicate  in  the  same  time  to  the  body  m,  impeded  by 
the  body  n,  by  the  proportion 


182  Dynamics 

m  —  n 

m  4-  n  '.  m  —  n  :  :  p-  : <r. 

^       m  -\-  n^ 

If,  therefore,  we  call  w  the  velocity  of  m  at   the  expiration  of  a 
264.    number  t  of  seconds,  we  shall  have 

m  —  n 

w  = £r  t; 

m  X  n  °     ' 

267.    and  the  space  which  it  will  have  described,  will  be 


m  —  n 
m 


which  is  readily  found,  by  putting  for  t  the  given  number  of  sec- 
276.    onds,  and  for  g  32,2  feet. 

293.  If  at  the  first  instant  the  body  n,  supposed  to  have  less 
mass  than  the  other,  receive  an  impulse  or  velocity  v,  that  is,  if  it 
were  struck  in  such  a  manner,  that,  being  considered  free  and  without 
gravity,  it  would  pass  over  in  a  second  a  number  of  feet  denoted 
by  V,  it  would  divide  this  action  with  the  body  m  which  it  would 
draw  during  a  certain  time.  In  order  to  determine  how  the 
action  in  question  would  be  divided,  it  must  be  remarked,  that 
at  the  first  instant  the  action  of  gravity  being  infinitely  small  or 
nothing,  the  body  n,  urged  with  a  velocity  v,  acts  upon  the  body 
m  as  if  this  last  were  at  rest.  It  is  necessary,  therefore,  in  order 
to  find  the  velocity  remaining  after  the  action,  to  divide  the 
28.     quantity  of  motion  n  v  hy  the  sum  of  the  masses,   which  gives 

— -J- —  for  the   velocity   with  which   n  would   draw  m,  if  gravity 

did  not  act  in  the  following  instants.  But  as  we  have  seen  that 
it  would  act  in  such  a  manner  as  to  give  to  the  body  m,  in  the 

opposite  direction,  the  velocity  — — —  g  t  in  the  time  t ;  it  fol- 
lows that,  at  the  expiration  of  the  time  t,  the  body  n  will  have 
only  the  velocity 

n  V  m  —  n 

m  -^  n  m  -\-  n  ^    ' 

Whence  it  will  be  seen,  that  however  small  n  may  be,  and  how- 
ever small  the  velocity  v,  and  however  considerable  the  mass  of  the 


Collision  of  unelastic  Bodies.  183 

body  m,  n  will  always  draw  m  for  a  certain  time,  after  which  m 
will  prevail,  and  draw  n  in  its  turn. 

Indeed,  whatever  may  be  the  quantity  of  motion  n  v,  impressed 
upon  n,  so  long  as  it  is  of  a  finite  value,  it  is  evident  that  it  would 
always  be  necessary,  in  order  to  counteract  it,  that  gravity  should 
act  for  a  certain  time,  for  it  only  acts  by  degrees  infinitely  small  at 
each  instant. 

If  we  would  know  at  the  expiration  of  what  time  m  will  cease 
to  ascend,  we  should  proceed  thus.  Let  t'  be  the  time  employed, 
by  a  heavy  body,  falling  freely,  in  acquiring  tlie  velocity  v ;  accord- 
ing to  article  263,  we  shall  have 

V  =gt'; 

therefore  the  velocity  of  n  will  be  changed  to 

n  ff  t'  m  —  n 

— - —    —     p- 1  • 

m  -\-  n  m  -^  n   °     ' 

which  being  put  equal  to  zero,  gives 

ngf    =    {m  —  n)gt, 

from  which  we  deduce 

nt' 
m  —  n' 

If,  for  example,  the  velocity  v,  supposed  to  be  impressed  upon  n 
is  such  as  a  heavy  body  would  acquire  in  one  second,  we  should 
have  i'  =  \".  Suppose  m  =  lOO"'-,  and  n  =  l"*-,  we  should 
have 

—     ^"     —  -H 

~    100-1    ~     99  ' 

that  is,  the  body  n  would  draw  the  body  m  only  during  one  ninety- 
ninth  of  a  second ;    still  it  would  draw  it. 

We  see,  therefore,  that  there  is  not  a  finite  force,  however 
small,  which  is  not  capable  of  overcoming  the  weight  of  a  body  ; 
and  that  it  is  not  possible  for  a  body  actually  in  motion,  to  be 
placed  in  equilibrium  with  the  weight  of  another  body,  that  is, 
with  a  body  that  has  the  simple  tendency  of  gravity.     The   former 


184  Dynamics. 

would  first  draw  the  latter,  and  afterward  be  drawn  by  it ;  there 
would  indeed  be  an  instant  of  rest,  but  it  would  be  that  in  which 
the  former  had  lost  all  the  velocity  impressed  upon  it,  and  this 
state  would  continue  only  for  an  instant. 

294.  Thus  the  force  of  bodies  in  motion  cannot  be  estimated 
by  weights,  that  is,  by  the  simple  tendency  of  gravity  in  bodies 
destitute  of  local  motion  ;  but  only  by  other  forces  of  the  same 
kind,  as  those  of  heavy  bodies  having  fallen  from  a  certain  height. 
Hence,  in  order  to  have  an  idea  of  the  force  of  a  body  of  3 
pounds,  carried  with  a  velocity  of  50  feet  in  a  second,  1  should 
seek  by  the  method  of  article  277,  from  what  height  a  heavy 
body  must  fall  to  acquire  a  velocity  of  50  feet  in  a  second,  and 
I  should  find  it  to  be  38,8  feet  nearly.  I  should  conclude,  there- 
fore, that  a  body  of  3  pounds,  urged  with  a  velocity  of  50  feet  in 
a  second,  must  strike  as  if  it  had  fallen  from  a  height  of  38,8 
feet. 

295.  The  force  which  bodies  in  motion  are  capable  of  exerting, 
is  called  percussion. 

The  force  of  percussion  cannot,  therefore,  in  any  way  be  com- 
pared with  simple  pressure,  or  the  effort  which  a  mass  is  capable 
of  making  by  its  weight  without  local  motion.  A  blow  of  a 
hammer,  though  feeble,  will  driv^e  a  nail  into  a  block  of  wood  ;  also 
a  body  of  small  mass,  which  by  its  fall  had  acquired  but  litde 
velocity,  would  be  attended  with  the  same  result,  while  a  very 
considerable  weight  would  produce  no  effect. 

The  reason  of  this  difference  is,  that  in  the  former  case,  all 
the  degrees  of  velocity  possessed  by  the  body  in  motion,  are  exert- 
ed in  an  instant ;  whereas  in  the  latter,  the  weight,  which  exerts 
only  a  pressure,  receives  its  degrees  of  force  successively,  and 
imparts  them  in  the  same  manner  to  the  nail  and  the  surrounding 
mass  ;  and  as  each  of  these  degrees  is  infinitely  small,  it  is  ab- 
sorbed as  soon  as  it  is  received. 


Collision  of  elastic  Bodies.  185 


Of  the  Collision  of  elastic  Bodies. 

296.  Althoui^h  elastic  bodies,  according  to  the  definition  which 
we  have  given,  must  be  compressible,  we  are  not  hence  to  infer  that 
they  must  be  so  much  the  more  compressible,  as  they  are  more 
elastic.  A  ball  of  wool,  for  example,  is  not  4nore  elastic  than  a 
ball  of  ivory,  although  it  is  much  more  coinpressible. 

Be  this  as  it  may,  compressibility  seems  to  be  inseparable 
from  elasticity.  A  body  in  virtue  of  its  compressibility,  changes 
its  figure,  when  a  force  is  applied  to  it  from  without;  and  in  vir- 
tue of  its  elasticity,  it  tends  to  recover  this  figure.  But  among 
all  elastic  bodies,  some  recover  their  figure  entirely,  others  only 
in  part.  These  last  are  called  imperfectly  elastic  bodies.  As  to 
the  former,  they  may  resume  their  figure  more  or  less  promptly, 
and  by  very  different  degrees.  But  if  they  are  such  that,  after 
being  struck,  they  restore  themselves  according  to  the  same  de- 
grees by  which  they  were  compressed,  we  call  them  perfectly 
elastic  bodies.  In  other  cases  they  are  denominated  simply  elastic 
bodies.  We  shall  here  consider  only  those  that  are  perfectly 
elastic. 

We  observe  with  respect  to  perfectly  elastic  bodies,  that  in 
collision,  a  resistance  takes  place  on  the  part  of  the  body  which 
has  the  least  velocity,  and  consequently  a  compression,  and  that 
on  this  account  not  only  a  restoration  of  the  figure  follows  this 
con)pression,  but  this  restoration  is  itself  followed  by  a  new 
change  of  figure  directly  contrary  to  the  first.  To  this  succeeds 
another,  which  reduces  the  body  to  the  figure  first  given  by  the 
compression,  and  so  on.  In  this  way  the  parts  of  each  body 
have,  with  respect  to  their  centre  of  gravity,  a  vibration,  or  mo- 
tion backward  and  forward  ;  since  the  parts  tend  to  return  to 
their  first  figure  by  a  motion  which  goes  on  increasing,  and  thus 
carries  them  beyond  their  former  position.  These  changes  of 
figure,  which  alternate  with  each  other,  are  sensible  in  several 
elastic  bodies,  when  struck,  and  particularly  in  those  that  are  of 
the  sonorous  class. 

Mech.  24 


134. 


186  Dynamics. 

It  is  not,  however,  to  be  supposed,  that  these  vibrations  affect 
the  velocity  which  the  bodies  take  after  collision.  Tliey  can 
have  no  influence  upon  the  motion  of  the  centre  of  gravity,  since 
this  motion  takes  place  in  each  of  the  two  bodies  independently 
of  the  other. 

The  collision  of  perfectly  elastic  bodies  is  to  be  viewed, 
therefore,  in  the  following  manner.  When  the  two  bodies  m,  n, 
Fig.148.  come  to  meet  in  C,  the  resistance  which  n  opposes  to  w,  causes 
them  to  be  mutually  compressed,  until  the  two  centres  and  the 
point  of  contact  have  all  the  same  velocity  ;  thus  far  every  thing 
takes  place,  as  in  the  ctjllision  of  hard  bodies,  with  the  exception 
of  the  change  of  figure,  which  can  contribute  nothing  to  the 
quantity  of  motion  lost  or  gained. 

The  change  of  figure  is  effected  in  such  a  manner,  that  each 
of  the  two  bodies  is  flattened  to  the  same  degree  on  opposite 
sides;  since  the  j)arts  farthest  removed  from  contact,  advancing 
more  rapidly  in  the  one  body  and  less  rapidly  in  the  other, 
until  the  compression  is  com[)leted,  crowd  very  much  tlie  inter- 
mediate [)arts.  The  compression  once  finished,  the  parts  of 
each  body  bordering  upon  the  points  of  contact,  support  them- 
selves the  one  against  the  other,  while  the  contact  is  transferred  ; 
and  the  recoil  of  the  spring  takes  place  toward  the  parts  oppo- 
site to  the  point  of  contact,  with  all  the  force  by  which  the  bodies 
tend  to  restore  their  figure. 

It  will  accordingly  be  seen  that  the  impinging  body  loses  by 
the  recoil,  a  velocity  equal  to  that  which  it  iiad  lost  by  the  com- 
pression ;  and  that,  on  the  other  hand,  the  impinged  body  gains 
by  the  recoil  a  velocity  equal  to  that  which  it  had  gained  dur- 
ing the  compression;  and,  although  the  two  bodies  do  not  cease 
to  exert  their  elastic  force  when  they  have  regained  their  origi- 
nal figure,  they  have  no  longer  any  action  upon  each  other, 
since,  the  force  with  which  they  go  on  to  dilate  themselves  be- 
ginning now  to  grow  less,  they  separate  from  each  other  at  this 
conjuncture. 

If,  when  the  two  bodies  move  in  the  same  direction,  u  is  the 
velocity  of  the  impinging  body,  and  v  that  of  the  impinged  ;  u' 
being   supposed    the    common    velocity    which    they    would    have 


Collision  of  elastic  Bodies.  187 

after  collision,  considered  as  unelastic,  u  —  u'  would  be  the 
velocity  lost  by  the  impinging  body  ;  when,  therefore,  the  recoil 
of  the  spring  (taking  place  in  a  direction  opposite  to  the  motion) 
causes  as  much  motion  to  be  lost,  as  had  already  been  lost  by  the 
compression,  there  will  remain  only  the  velocity 

u  —  2  [it  —  u')  =  2  u'  —  u. 

As  to  the  impinged  body,  u'  —  v  is  the  velocity  gained  by  col- 
lision ;  and  we  have  seen  that,  by  the  recoil  of  the  spring,  it  ac- 
quires as  much  more  ;  it  will  have,  therefore, 

V  -j-  2  (w/  —  v)  :=  2  u'  —  V. 

This  case  comprehends  that  in  which  one  of  the  tw'O  bodies  is  at 
rest  before  collision. 

If  the  bodies  move  in  opposite  directions,  the  reasoning  is  pre- 
cisely the  same  for  the  one  which  has  the  greater  quantity  of 
motion.  As  to  the  other,  it  would,  considered  as  unelastic, 
lose  its  velocity  by  the  collision,  and  acquire  another  in  the  op- 
posite direction  ',  u'  being  this  velocity,  we  shall  have  v  -\-  u' 
for  the  velocity  lost.  Doubling  this  effect,  on  account  of  the 
bodies  being  elastic,  and  adding  it  to  the  original  velocity  —  v^  we 
have 

2  (r  +  «')  —  V  —  2  m'  +  V. 

297.  By  attending  to  the  resulting  expression  in  each  of  the 
above  cases,  it  will  be  seen  that  the  circumstances  of  the  collis- 
ion of  bodies  perfectly  elastic  are  all  comprehended  in  this  single 
rule  ; 

Seek  the  common  velocity  which  the  two  bodies  would  have  after 
collision,  if  they  tvere  destitute  nf  elasticity  ;  then  from  double  this 
velocity,  take  the  velocity  which  each  had  before  collision,  and  we 
shall  have  the  velocity  of  each  after  collision ;  it  being  understood 
that  when  the  bodies  move  in  opposite  directions  before  collision, 
the  sign  —  is  to  be  given  to  the  velocity  of  that  body  which  has  the 
less  quantity  of  motion. 

298.  From  the  principles  above  laid  down,  we  might  easily 
obtain,  for  the  collision  of  elastic  bodies,  formulas  which  should 
contain  only   the  masses   and  velocities  before  collision.     In  order 


1 88  Dynamics. 

to  this,  it  would  only  be  necessary  to  substitute,  in  the  expressions 
2  w'  —  u  and  2  u'  ±  v,  the  value  of  u'  furnished  by  die  rules 
of  articles  286,  288.  But  as  these  formulas  would  not  present 
themselves  in  a  manner  so  easy  to  be  retained  as  the  rules  we  have 
given,  we  leave  this  substitution  to  be  made  by  those  who  may  wish 
to  see  the  result. 

299.  We  observe  that,  when  one  of  the  two  bodies  is  at  rest, 
the  velocity  which  it  would  receive  by  the  collision  is  double  that 
which  it  would  have  had,  considered  as  non-elastic.  This  is  an  evi- 
dent consequence  of  the  general  rule. 

300.  To  give  a  few  examples  of  these  rules,  let  us  suppose,  in 
the  first  place,   that  the  two   bodies   are   equal,   and  that  one  of 

them  is  at  rest ;  then  '-, — ,  which  expresses  the  velocity   after 

m  -\-  n  '  -^ 

Til  u 

collision,  the  bodies  beins;  considered   as  unelastic,  becomes  ^ — '-  or 

297.  I  u.  From  twice  -i-  m  or  u,  therefore,  we  subtract  u  to  obtain  the 
velocity  of  the  impinging  body  after  collision,  which  is  consequently 
zero.  To  find  the  velocity  of  die  impinged  body,  from  twice  \  u 
or  u,  we  subtract  0,  which  it  had  before  collision,  and  we  have  u 
for  the  velocity  after  collision.  Hence  we  see  that  the  motion  of 
the  impinging  body  passes  wholly  into  the  impinged.  Accord- 
ingly, if  several  equal  elastic  bodies  be  placed  in  contact  with  each 
other  in  the  same  straight  line,  and  one  of  them  be  made  to  im- 
pinge against  the  others  in  the  direction  of  this  line ;  the  only  effect 
would  be,  that  the  one  at  the  opposite  extremity  would  be  driven 
off  with  the  same  velocity.  If  two  are  made  to  impinge  at  the 
same  time  against  the  others,  two  would  be  detached  from  the  other 
extremity,  and  so  on. 

Let  us  suppose  the  two  bodies  to  move  in  the  same  direction, 
one  of  5  ounces,  and  with  a  velocity  of  6  feet  in  a  second,  and 
the  other  of  7  ounces,  with  a  velocity  of  2  feet  in  a  second.  For 
the  common  ^velocity  which  they  would  have  after  collision,  consid- 
ered as  unelastic,  we  obtain 

5X6  +  7X2  _  ,^  _  _ 
5  4-7  —12—^3- 

If,  therefore,  from  double  this  quantity  or  7^,  we  take  the  veloci- 
ties before  collision,  namely,  6  and  2  respectively,  we  shall  have 


Collision  of  elastic  Bodies.  IS^ 

for  the  velocity  of  the  impinging  body  after  collision  1|,  and  for 
that  of  the  impinged  5i. 

If  the  impinged  body,  instead  of  7  ounces,  had  a  mass  of  20 
ounces ;  the  velocity,  after  collision,  the  bodies  being  considered 
as  unelastic,  would  be, 

5  X  6  +  20  X  2  _  ^„  _ 

5  +  20  —  25   —  ^5- 

If  from  double  this  quantity  or  5|,  we  subtract  the  velocities  be- 
fore collision,  namely,  6  and  2  respectively,  we  shall  have, 

5|  — 6     and     5f  —  2, 
that  is 

—  I     and     3|, 

for  the  velocities  after  collision,  in  which  the  sign  —  before  |  indi- 
cates that  the  impinging  body  would  rebound. 

If  the  two  bodies  are  made  to  move  in  opposite  directions  with 
the  same  masses  and  the  same  velocities,  as  in  the  first  of  the 
above  examples,  the  velocity  after  collision,  the  bodies  being  con- 
sidered as  unelastic,  would  be  289, 

5  X  6  —  7  X  2  _  30  —  14  _  ,_,  _  ,^ 

5-1-7  —        12        —  12  —  ^3- 

If  from  double  this  velocity  or  2|,  we  subtract  the  velocity  6, 
which  the  impinging  body  had  before  collision,  we  shall  have 
—  3i  for  its  velocity  after  collision;  it  will  rebound,  therefore,  with 
a  velocity  of  3i  feet.  As  to  the  impinged  body,  it  will  be  recol- 
lected that  to  twice  11  or  2|,  the  velocity  before  collision  is  to  be 
added,  which  gives  4|  for  its  velocity  after  collision.  297. 

301.  Since,  when  elastic  bodies  move  in  the  same  direction 
before  collision,  the  velocities  after  collision  are  297. 

2  u'  —  u     and     2  u'  —  v, 

u'  being  the  velocity  which  they  would  have,  considered  as  une- 
lastic ;  the  difference  u  —  v  o(  these  two  velocities,  is  the  same 
as  the  difference  of  the  velocities  before  collision.  This  difference 
is  called  the  relative  velocity,  and  is  accordingly  the  same  before 
and  after  collision. 


397. 


190  Dynamics. 

When,  on  the  other  hand,  the  bodies  move   before   collision  in 
opposite  directions,  their  velocities  after  collision  are, 

2  ti'  —  u  and  2  u'  -\-  v, 
the  difference  of  which  is  u  -\-  v,  and  this  was  their  relative  velo- 
city, or  that  witli  which  they  approached  each  other  before  col- 
lision. Therefore  the  velocity  with  which  they  separate  from 
each  other  after  collision,  is  the  same  as  that  with  which  tliey 
approach  each  other  before  collision ;  thus,  with  respect  to  elastic 
bodies  the  relative  velocity  is  the  same  before  and  after  collision. 


Of  the  Motion  of  Projectiles. 

302.  By  the  motion  of  projectiles,  we  understand  that  of 
bodies,  which,  being  thrown  with  a  certain  force,  are  afterward  left 
to  the  action  of  this  force  and  that  of  gravity.  We  shall  first  seek 
the  path  that  would  be  described  in  free  space. 

^".149.  From  the  point  A,  let  a  body  be  thrown  in  the  direction  AZ, 
and  with  any  given  velocity.  If  gravity  were  out  of  the  ques- 
tion, it  would  move  uniformly  in  the  direction  of  the  straight  line 
AZ.  But  as  gravity  acts  without  interruption,  the  body  will  not 
be  in  the  straight  line  AZ,  except  for  an  instant ;  instead  of  AZ^ 
it  will  describe  a  curved  line  ABC  o{  which  AZ  will  be  the  tan- 

Q  gent  at  the  point  A,  since  AZ  is   one  of  the  instantaneous  direc- 

97.        lions  of  the  moving  body. 

303.  In  order  to  determine  the  nature  of  this  curve,  let  AE 
be  the  velocity  communicated  to  the  projectile,  or  the  number  of 
feet  that  it  would  describe  in  a  second,  if  it  preserved  continually 
this  velocity ;  and  at  the  instant  of  its  leaving  the  point  A,  let 
us  suppose  tliis  velocity  composed  of  two  others,  one  AD  hori- 
zontal, and  the  other  AF  in  a  vertical  direction.  It  is  evident 
that  the  direction  of  gravity  being  vertical  or  perpendicular  to 
AD,  its  action  will  not  tend  either  to  diminish  or  increase  the  ve- 
locity AD,  and  that  consequently  whatever  course  the  body  may 
take,  it  will  preserve  constantly  the  same  velocity  parallel  to  the 
horizon. 

As  to  the  velocity  in  the   direction   AF,   when   the    body,    in 
virtue  of  its  constant  velocity,  parallel  to  the  horizon,  shall  have 


Motion  of  Projectiles.  191 

advarxed  by  a  quantity  equal  to  AP,  it  will  not  have  risen  to 
a  height  P±i,  equal  to  that  at  which  it  would  have  arrived, 
uninfluenced  by  gravity,  but  to  some  lower  point  M  in  the  same 
vertical  PN;  because,  its  velocity  in  a  vertical  direction  being 
directly  opposed  to  iliat  of  gravity,  tlie  space  which  it  would 
have  described  in  virtue  of  this  vertical  velocity,  must  be  dimin- 
ished by  the  space  which  the  action  of  gravity  would  have  caused 
the  body  to  describe  in  the  same  time. 

Accordingly  let  v  denote  the  velocity  communicated  in  the 
direction  AZ  or  the  number  of  feet  that  the  projectile  would  de- 
scribe uniformly  each  second,  in  virtue  of  this  velocity,  and  t  the 
time,  or  number  of  seconds  or  parts  of  a  second,  employed  in  pass- 
ing from  A  to  some  point  N,  we  shall  have  263. 

AN  =  V  t. 

Let  g  be  the  velocity  communicated  by  gravity  in  a  second, 
^  g  t^  will  be  the  space  that  a  heavy  body  would  describe  in  a 
number  t  of  seconds.  If  therefore  M  be  the  point  where  the  body 
will  arrive  at  the  expiration  of  the  time  t,  we  shall  have 

NM  =  ig  t"^.  275. 

Through  the  point  A,  draw  the  vertical  AX,  and  through  the 
point  Mthe  straight  line  MQ  parallel  to  the  tangent  AZ.  Calling 
AQ,  x',  and  QM,  which  is  equal  to  AN,  y',  we  shall  have 

x'  :=  \  g  t^,  and  y'  =  v  i. 

If  from  this  last  equation  we  deduce  the  value  of  i,  namely, 

t-^ 

V 

and  substitute  it  in  the  first,  we  shall  obtain 


or 

But  ^  expresses  the  height  from  which  a  heavy  body  must   fall   to 
acquire  the  velocity  v ;  hence,  if  we  call  this  height  h,  we    shall     277. 


192  Dynamics. 


have  ^—  =  h,  and  consequently  j-^  z=i  4  h; 

therefore, 

4ha:'  =  y'^. 

We  hence  infer  that  each  point  M  of  the  curve  AMC  has  this 
property,  that  the  square  of  the  ordinate  y'  or  (^M,  j)arallel  to  the 
tangent  AZ,  is  equal  to  the  product  of  the   abscissa  ^Q  or   x'   by 

rp^j       a  constant  quantity  4  A;  therefore  the  curve   AMC  is   a   parabola 

176.  which  has  for  a  dianieter  the  vertical  line  AX.,  and  for  its  param- 
eter the  quadruple  of  the  height  due  to  the  velocity  of  projection, 
and  of  which   the  angle  AQ^M,    made   by   the  ordinates   with   this 

Trig,     diameter,  is   the   complement  of    the  angle   of   projection    ZAC. 

182.  'Yh'is  curve,  therefore,  is  easily  constructed,  when  the  velocity  of 
projection  and  the  angle  of  projection  are  known. 

304.  We  proceed  to  examine  some  of  the  properties  of  this 
curve,  considered  as  the  path  traced  by  a  projectile  ;  and  for  this 
purpose  we  refer  the  different  points  31  to  the  horizontal  line 
AC  by  drawing  PM  perpendicular  to  AC. 

We  designate  AP   by  x,  PM  by  y,   and  the  angle  of  projec- 
Trig.  30.  tJQi-,  ZAC   by  a.     In  the  right-angled  triangle  APN  we  have 

1   :  AN  ::  sin  NAP  :  PN, 
:  :  cos  NAP  :  AP ', 

whence 

PN  =  AN  sin  NAP  =  V  t  sin  a, 
and  AP  or  x  =:  v  t  cos  a. 

Also,  since  MN  =  \  g  t^,  as  we  have  seen  above, 
PM  or  y  =:  V  i  s'm  a  —  ^  g  t^. 
Deducing  from  the  former  equation  the  value  of  t,  namely, 

X 

t  = , 

V  cos  a 

and  subsdtuting  it  in  the  latter,  we  shall  have, 

X  sin  a  ^  ^  x^ 

"  cos  a  »2  (,Qg  0(2' 


Motion  of  Projectiles.  193 


or. 


v^  X  sin  a 


^  S  ^  g  cos  a       cos  a^' 

or,  putting  for  ^ —  its  value   4  ^,    and  multiplying  both   members 
by  cos  a^, 

4  h  y  cos  a^  =  4  7t  a:  sin  a  cos  a  —  a;^, 
which  will  furnish  us  with  the  following  properties. 

305.  As  the  velocity  communicated  to  the  projectile  is  sup- 
posed to  be  limited  to  a  certain  measure,  its  effect  in  a  vertical 
direction  must  be  exhausted  at  the  end  of  a  certain  time  by  the 
action  of  gravity,  so  that  at  a  certain  point  the  body  will  cease 
to  ascend,  and  thence  will  commence  a  downward  motion  ;  but, 
as  its  horizontal  velocity  does  not  change  when  it  has  reached 
its  highest  point,  as  B,  it  will  describe  the  second  branch  jBCof 
the  same  curve,  and  will  again  meet  the  horizontal  line  -<4C  in 
another  point  C.  Now  in  order  to  determine  the  distance  AC, 
called  the  horizontal  range  *  of  the  projectile,  we  have  only  to  sup- 
pose y  =i  0.     We  have,  accordingly, 

4  h  X  sin  a  cos  a  —  a:''  or  x  (4  h  sin  a  cos  a  —  x)  :=  0  ; 

which  gives  x  =  0,  and  x  =  4  h  sin  a  cos  a.  The  first  value  of 
X  indicates  the  point  A  ;  the  second  is  that  of  AC,  which  may  be 
determined  by  producing  XA  till  AK  is  equal  to  4  h,  and  letting 
fall  from  the  point  K  upon  AZ  the  perpendicular  KL,  and  from 
the  point  L  upon  AC  the  perpendicular  LC ;  since  we  have 

i?  =  1  :  sin  jfiT  =  sin  a  :  :  AK  =  4  h  :  AL  =  4  A  sin  a, 
and  R  =  I  :  sin  ALC  =.  cos  a  :  :  AL  =  4  h  sin  a  :  AC 
=  4  A  sin  a  cos  a. 

306.  If  with  the  same  velocity  of  projection  we  would  know 
what  angle  would  give  the  greatest  horizontal  range,  we  take  the 
differential  of  the  value  of  AC,  by  regarding  n  as  variable,  and 

put  this  differential  equal  to  zero  ;  thus  Cal.  45. 

4  h  d  a  cos  a^  —  4  A  c?  a  sin  a^  =  0  ; 

*  Sometimes  called  also  random  and  amplitude. 
Mech.  25 


194  Dynamics. 

from  which  we  deduce, 


or 


Trig.  8.  that  is, 

and  consequendy, 


sin  a^  ^ 

COS  a~ 


tang  a^  z=   1, 


tang  a  =:  1, 


in  other  words,  the  tangent  of  the  angle  of  projection  is  in  this 
case  equal  to  radius  j  accordingly  this  angle  is  equal  to  45°. 
'  Therefore,  the  greatest  horizontal  range  is  obtained,  other  things 
being  the  same,  when  the  angle  of  projection  is  45°.  It  is  here  sup- 
posed that 
'^"S-^O.  gj^  a  z=  cos  a  =  j^I ', 

this  value  substituted  in  the  above  expression  for  AC,  gives 

./4C  =  4  A  VI  VI  =  4  A  X  I  =  2  A; 

therefore,  the  greatest  random  is  double  the  height  through  which  a 
body  must  fall  to  acquire  the  velocity  of  projection. 

307.  If  we  would  know  to  what  height  the  body  ascends,  or 
the  highest  point  B  of  the  curve,  we  proceed  thus  ;  in  the  equa- 
tion 

4  hy  cos  a^  =z   Ah  x  sin  a  cos  a  —  o^', 

we  put  equal  to  zero,  the  differential  of  y,  taken  by  regarding  x 
only  as  variable,  which  gives 

A  h  d  X  sm  a  cos  a  —  2  x  d  x  =.  0, 

from  which  we  obtain 

X  =^  2  h  sin  a  cos  a ; 

therefore,  since  -4C  =  4  A  sin  c  cos  a,  if  we  suppose  the  perpen- 
dicular BD,  we  shall  have 

X  or  AD  =  2  h  sin  a  cos  a  =  I  AC. 

Moreover,  this  value  of  x  being  substituted  in  the  equation, 


Motion  of  Projectiles.  195 

4  h  y  cos  a^   =   4  A  a;  sin  a  cos  a  —  x^, 
gives 

4  A  y  cos  a^   =    8  A^  sin  a^  cos  a^  —  4  A^  sin  a*  cos  a% 

from  which  we  obtain 

y  or  BD  =:  h  sin  a\ 

This  determines  the  vertex  of  the  axis,  since  d  y  being  zero  at  the 
point  B,  the  tangent  at  B  is  parallel  to  AC,  or  perpendicular  to 
BD. 

308.  We  propose  now  to  determine  the  direction  AZ,  to  be 
given  to  a  projectile  in  order  that  it  may  fall  upon  a  known  point  Fig.l50. 
M,  that  is,  the  inclination  that  a  mortar,  for  instance,  must  have  to 
throw  a  shell  upon  the  known  point  M. 

The  perpendicular  MP  upon  the  horizontal  line  passing  through 
the  point  A,  being  drawn,  the  distance  AP,  and  the  angle  MAP, 
are  to  be  considered  as  known.  AP  being  designated  by  c,  and 
the  angle  MAP  by  e,  we  shall  have 

Ti/trt  c  sin  c 

cos  e  :  c  :  :  sm  e  :  JVijr    =   , 

cos  e 

we  have,  therefore,  for  the  point  M,  x  =  c,  and 

c  sin  e 
•^  cos  e 

Substituting  these  values  in  the  equation 

4  A  y  cos  a^   =   4  h  x  sin  a  cos  a  —  a;^, 

we  obtain, 

4  A  c  sin  e  cos  a^ 


cos  e 
or 


=:    4  A  c  sin  «  cos  a  —  c^. 


4  A  sin  e  cos  a^   =   4  A  sin  a  cos  a  cos  e  —  c  cos  e, 

or 

4  A  cos  a  (sin  a  cos  e  —  sin  e  cos  a)    =   c  cos  e, 

that  is,  "^"S.". 

4  A  cos  a  sin  (a  —  e)    =   c  cos  c ; 


196  Dynamics, 

or,  since 

Trig.  27.  cos  a  sin  (a  —  e)  =  |  (sin  {a  -{-  a  —  e)  —  sin  (a  —  a  +  e)  ), 
4  A  i  (sin  (2  a  —  e)  —  sin  e)    =    c  cos  e, 


and 


2  A     .     /^  V  2  A  sin  c    , 

sin  (2  rt  —  e)   = \-  c, 


COS  e         ^  COS  e 

which  may  be  given  by  the  following  construction. 

Having  raised  upon  AM  the  indefinite  perpendicular  AE; 
from  the  middle  D  of  AK  =  4  h,  we  erect  upon  AK  the  perpen- 
dicular DE,  cutung  AE  in  some  point  E,  from  which  as  a  centre, 
and  with  a  radius  equal  to  EA,  we  describe  the  arc  ANN'K; 
having  produced  PM  till  it  meets  this  arc  in  the  points  iV,  N',  if 
we  draw  the  lines  ANZ,  AN'Z',  these  will  be  the  directions  in 
which,  the  projectile  being  thrown  with  a  velocity  due  to  the  height  A, 
it  will  in  either  case  fall  upon  the  point  M. 

Geom.  Indeed  it  will  be  readily  seen  that  the  angle  EAD   of  the 

right-angled  triangle   ADE  is  equal  to  MAP.     Therefore,  since 

AD  =  2  A,  ED  = :  and,  since  AP  =.  c,  we  shall  have, 

cos  e 

ED  +  AP   or   El  =  ?±!i!L?  +  . 

cos  e  ' 


consequently, 

2  A 


cos  e 


sin  (2  a  —  e)  =  EL 


But  in  the  same  triangle  ADE,  AE  =.  ;    therefore 

°  cos  e 

AE  sin  {2  a  — e)   =    EL 

Let  the  arc  KNA  be  produced  till  it  meets,  in  G,  the  vertical  GE, 
and  from  the  points  N,  N',  draw  the  perpendiculars  NL,  N'L'. 
In  the  triangle  NEL,  we  have 

NE   :    NL,  or  AE    :    EL  ::    I    :    sm  NEG', 
whence, 

AE  sin  NEG  =  EL-, 

accordingly. 


Motion  of  Projectiles. 
sin  (2  a  —  e)    =  sin  JVEG, 


197 


and 


2a 


e   =  JVEG  =  JVEA  +  e; 

consequently, 

a    =    1  JVEA  +  e. 

But  because  the  angle  JVAM  has  its  vertex  in  the  circumfer-  q^^^ 
ence,  and  AM  is  a  tangent,  JVAM  is  equal  to    |  JYEA  ;  also  the  I3i. 
angle  MAP  =  e  j  whence 

a  =  JV./3  Jf  +  MAP  =  JVAP ; 

therefore  the  point  JV  satisfies  the  question. 

The  same  may  be  shown  with  respect  to  the  point  JV'.  Indeed 
in  the  triangle  JV'ED,  we  have 

JV'E    :    N'L',    or    AE    :   EI   ::    1    :    sin  JV'ED, 

:  :    1    :   sin  JV'EG, 

whence 

AE  sin  JV'EG    =    EI', 

and,  since 

AE  sin  {2  a  —  e)   =  El, 
as  above  shown,  we  have 

sin  (2  a  —  e)  =  sin  KEG, 

and 

2  a  —  e  =  A'-'EG  =  N'EA  +  e  ; 

therefore, 

a  =  1  JVjE^  +  e  =  KAM-^  MAP  =  A^'v^P. 

309.  Thus  with  the  same  force  of  projection,  a  projectile  may 
always  be  made  to  fall  upon  the  same  point  M,  according  to 
two  different  directions,  provided  that  AP  does  not  exceed  DR. 
The  direction  AN'  is  the  most  favorable  for  crushing  buildings 
or  other  objects  with  shells.  The  direction  AN  is  to  be  pre- 
ferred, when  the  purpose  is  simply  to  throw  down  walls  and 
breast-works,  and  by  rebounding  to  lay  waste  at  a  distance.  This 
leads  us  to  speak  of  ricochet  firing ;  but  we  shall  first  remark 
that  the  equation  x  =:  v  t  cos  a,  found  above,  gives  a  simple  ex- 


198  Dynamics. 

pression  for  the  time  employed  in  passing  from  Jl  to  any  point 
M.     We  have  only  to  put  for  x  its  value  c,   and   for   v  its  value 
277.    \/ 2gh,  and  we  have 


cos  a  y/2gh 


Now  we  have  seen  how  h  is  determined  by  experiment,  and  we 
know  that  g  =  32,2  feet. 

Trig.  20.        When  the  inclination  is  45°,  cos  a  being  -y^T  or  \  s/\  we  have 


t  = 


2- 
C 


J^/2^/2^A  s/ gh 


hence,  if  the  point  M  is  in  a  horizontal  line,  which  gives  c  equal 
306.     to  the  range  or  to  2  A,  we  obtain 


t=   ^* 


\^g  -v/A 


-J- 


This  general  expression  for  the  time  may  be  made  use  of  in  regu- 
lating the  fusees  of  bombs.  We  proceed  now  to  ihe  subject  of 
ricochet  firing. 

310.  By  the  above  term  is  meant  a  motion  by  which  a  pro- 
jectile, after  meeting  with  an  obstacle,  rebounds  and  commences 
a  new  motion  similar  to  the  first.  The  smaller  the  angle  of  ele- 
vation above  the  horizon,  the  greater,  other  things  being  the  same, 
is  the  tendency,  upon  rebounding,  to  proceed  forward ;  since  the 
projectile  force  is  exerted  almost  entirely  in  a  horizontal  direc- 
tion, and  much  time  is  required  for  the  resistance  of  the  air  and 
other  obstacles  to  destroy  it.  If  the  projectile  be  of  an  unelastic 
substance,  and  the  surface  upon  which  it  falls  be  horizontal  and 
Fig.l5l.  unyielding,  it  would  not  bound,  since  upon  arriving  at  C,  accord- 
ing to  any  direction  Jk/C,  its  velocity  might  be  decomposed  into 
two  others,  of  which  (^C^  perpendicular  to  the  surface,  would  be 
simply  destroyed,  the  rebounding  in  other  cases  being  caused  en- 
tirely by  the  elasticity,  so  that  the  other  part  PC  remains  (no  ac- 
count being  taken  of  friction  and  the  resistance  of  the  air),  and  the 
body  would  move  along  CZ. 


Motion  of  Projectiles.  199 

311.  But  if  at  the  point  C,  where  the  body  meets  the  surface  Fig.  152. 
there   be  a  mound  or  eminence  CE,  the  motion  according  to  MC 
being  dpcomposed  into  two  otliers,  one  according  to  ^C  perpendic- 
ular to  the  surface  CE,  and  the  other  PC  in  the  direction  of  this 
surface,  the  body  would  proceed  according  to  this  latter,  describing 

the  line  PE,  and  might,  after  leaving  the  point  E,  describe  just 
such  a  curve  as  it  would  have  described,  if  it  had  been  projected 
from  E,  according  to  CE,  with  the  same  velocity  ;  so  that  it  would 
elevate  itself  to  a  certain  point,  and  then  return  to  the  surface  in 
some  other  point  /,  when  the  motion  under  similar  circumstances 
might  be  again  renewed. 

312.  A  ricochet  motion,  therefore,  depends  upon  the  position 
of  the  obstacle  against  which  the  body  in  question  strikes.  But  if 
the  obstacle  be  flexible  or  yielding  like  the  earth,  water,  &c.,  this 
motion  may  take  place  even  when  the  surface  is  perfectly  horizon- 
tal. Indeed,  by  the  vertical  velocity  QC,  the  body  tends  to  bury  Fig.  153. 
itself,  and   does  bury  itself  more  or  less,  according  to  the  nature 

of  the  obstacle  ;  while  with  the  velocity  PC  it  ploughs  the  earth, 
and  forms  a  furrow,  the  depth  of  which  increases  till  the  vertical 
velocity  QC  is  destroyed.  Then  by  the  remaining  velocity  in  a 
horizontal  direction,  it  drives  before  it  the  matter  which  lies  in 
its  way,  and  in  working  for  itself  a  passage,  it  inclines  in  the  direc- 
tion from  which  it  experiences  the  least  resistance,  and  the 
surface  of  the  furrow  becomes,  with  respect  to  the  body,  what 
CE  was  in  the  last  case.  Now  as  the  remaining  projectile  Fig.l52. 
force,  other  things  being  the  same,  is  so  much  the  greater  ac- 
cording as  the  depth  of  the  furrow  is  less,  and  as  this  depth 
depends  upon  the  vertical  velocity  QC  which  will  be  so  much  the 
less,  according  as  the  angle  MCP,  or  the  angle  of  projection  RAZ, 
is  less,  it  will  be  seen  how  the  smallness  of  the  angle  of  projection 
is  favorable  to  this  sort  of  motion. 

313.  The  figure  of  the  body  also  is  of  great  importance. 
If,  for  example,  the  question  related  to  a  motion  upon  water,  and 
the  body  were  of  a  spherical  shape,  the  velocity  MC  must  be 
such  that  the  vertical  velocity  QC  may  be  destroyed  before  the 
vertical  diameter  of  the  body  is  entirely  immersed,  since,  when 
the  body  is  once  covered,  the  resistance  of  the  water  would  act 
equally  in  every  direction,  and  there  would  be  nothing  to  change 


200  Dynamics. 

its  direction,  except  gravity,  the  tendency  of  which  would  be  to  pre- 
vent a  ricochet. 

314.  As  this  immersion,  however,  takes  place  gradually,  it 
will  be  seen  that  the  motion  of  the  centre  must  be  in  a  curved  line ; 

Fig.  154.  since  while  any  part  of  the  body  remains  above  the  surface, 
the  direction  in  which  the  resistance  acts  is  changing  continually. 
If,  for  instance,  when  the  centre  C,  after  having  described  any 
line  PC,  tends  to  move  according  to  the  prolongation  CI, 
of  this  line,  we  imagine  two  tangents  BR,  DS,  parallel  to  this 
direction,  it  is  evident  that  the  part  BVL  only  would  be  exposed 
to  this  resistance  ;  and  that  if  the  body  is  spherical,  the  resultant 
CK o(  all  the  resistances  exerted  upon  the  different  points  of  jBf^L 
would  have  a  direction  tending  to  elevate  the  body  above  CI',  so 
that  the  parallelogram  CIEK  being  formed,  CE  will  be  the 
course  which  the  body  would  take  instead  of  CI,  no  allowance 
being  made  for  gravity. 

315.  Finally,  if  the  body  and  the  obstacle  are  flexible  and 
elastic,  this  circumstance  will  further  contribute  to  a  ricochet 
motion.     We  take   a  very   simple   case,   as   an   example  ;  let  the 

Fig.155.  body  only  be  considered  as  flexible  and  elastic,  and  let  this  elas- 
ticity be  perfect ;  the  body  being  supposed  at  the  same  time  to  be 
destitute  of  gravity.  At  the  instant  in  which  the  body,  projected 
according  to  AC,  comes  to  touch  the  surface,  its  velocity  is 
decomposed  into  a  horizontal  velocity  which  would  remain  al- 
ways the  same,  if  there  were  no  friction,  and  no  resistance  on  the 
part  of  the  medium  in  which  the  body  moves.  As  to  the  perpen- 
dicular or  vertical  velocity  PC,  it  compresses  the  body,  and 
being  destroyed  gradually,  while  the  horizontal  velocity  continues, 
it  is  evident  that  the  centre  C  approaches  the  plane  HZ  by 
degrees,  which  go  on  decreasing,  while  the  rate  at  which  it  advan- 
ces parallel  to  HZ,  remains  the  same.  Consequently,  if  at 
each  instant  we  imagine  a  parallelogram  having  its  horizontal  sides 
to  its  vertical,  as  the  horizontal  velocity  is  to  the  velocity  that 
remains  in  a  vertical  direction,  the  diagonal  of  this  parallelogram, 
which  must  mark  the  course  of  the  centre  each  instant,  will 
be  different  and  differently  situated  each  instant,  so  that  the 
centre  C  will  approach  HZ  in  a  curve,  while  the  compression  is 
going  CD.     When  the  compression  has  ceased,  the  centre  C  will 


Motion  of  Projectiles.  201 

be  carried  for  an  instant  in  the  direction  of  a  tangent  parallel  to 
HZ;  after  which,  the  recoil  taking  place,  the  body  recovers  by 
degrees  the  velocity  by  which  it  tends  to  depart  from  the  plane 
after  the  same  manner  in  which  the  velocity  was  destroyed  by 
the  compression  during  its  approach  to  the  plane,  and  it  will  de- 
scribe the  second  part  RO  of  the  curve  perfectly  similar  to  RC. 
Lastly,  when  it  shall  have  arrived  at  the  point  O,  distant  from 
the  plane  HZ  by  a  quantity  equal  to  the  radius  IC,  it  will  move 
according  to  the  tangent  OT,  situated  like  AC;  that  is,  the  ob- 
lique collision  of  a  body  against  an  inflexible  and  unelastic  plane 
(friction  being  out  of  the  question)  lakes  place  in  such  a  manner 
as  to  make  the  angle  of  reflection  equal  to  the  angle  of  incidence, 
these  angles  having  for  their  measure  the  inclination  to  a  hori- 
zontal plane  of  the  tangents  at  the  extremities  C,  O,  of  the  curve 
described  by  the  centre  of  the  body  during  its  compression  and 
subsequent  recoil. 

316.  If  BD  be  the  direction  in  which  a  body  is  thrown,  re- 
gard being  had  to  gravity,  this  body  will  describe  the  portion 
DC  of  a  parabola  of  which  BD  is  the  tangent,  until  it  touches 
the  plane,  then,  when  the  compression  has  ceased,  it  will  describe 
another  portion  SO  of  a  parabola  equal  to  the  first  and  placed  in 
the  same  manner. 

317.  Friction,  moreover,  contributes  to  the  kind  of  motion 
under  consideration,  since  it  occasions  a  rotation  in  the  body  that 
aids  it  in  rising  above  obstacles,  as  we  have  already  seen.  234. 

318.  We  conclude  what  we  have  to  say  on  the  subject  of 
projectiles  moving  in  an  unresisting  medium,  with  observing  that, 
since  gravity  draws  a  body  downward  from  the  direction  given  it 
by  the  projectile  force,  when  we  take  aim  at  an  object  in  shooting 
or  in  throwing  any  body,  we  should  direct  the  sight  above  this 
object,  and  so  much  the  more  above  it,  according  as  it  is  more 
distant,  and  according  also  to  the  feebleness  of  the  force  employed. 
It  is  on  this  account  that  in  fire-arms  the  line  of  sight  makes 
an  angle  with  the  axis  of  the  piece,  so  that  these  lines  produced 
would  meet  at  a  point  beyond  the  muzzle  toward  the  mark. 
The  projectile,  ball,  or  bullet,  propelled  in  the  direction  of  the 
axis,  commences  its  motion  in  a  direction  making  a  greater  angle 
with  the  horizon  than  that  made  by  the   line   of  sight ;  so   that  the 

Mech.  26 


202  Dynamics. 

precaution  is  the  same  as  if  we  had  taken  aim   in   the   direction  of 
tiie  axis   but  at  a  point  above  the  object. 

319.  We  remark  further,  that  there  are  cases  in  which,  al- 
though we  have  given  no  impulse  to  a  body,  and  seem  to  aban- 
don it  to  gravity  alone,  yet  tiiis  body  describes  a  curved  line 
common  to  all  projeciiles.  A  body,  for  example,  which  is  suf- 
fered to  fall  from  the  mast-head  of  a  vessel  under  sail,  really  des- 
cribes a  curved  line.  If  we  attend  to  the  point  of  the  deck  where 
it  strikes,  we  shall  find  it  just  as  far  from  the  mast,  other  things 
being  the  same,  as  the  point  from  which  it  started,  so  that  the 
body  describes  a  line  parallel  to  the  mast ;  but  with  respect  to  a 
spectator  at  rest,  it  has  actually  described  a  parabola  (the  resist- 
ance of  the  air  not  being  considered),  for,  at  the  instant  it  w^as 
dropped,  it  must  have  had  the  same  velocity  with  the  vessel ;  the 
case  is  therefore  precisely  the  same,  as  if,  the  vessel  being  sta- 
tionary, we  had  thrown  it  with  a  velocity  equal  to  that  of  the 
vessel,  and  in  the  same  direction.  It  will  be  seen,  also,  at  the 
same  time,  why  it  describes  with  respect  to  the  mast  a  straight 
line  parallel  to  this  mast ;  it  is  because  they  both  move  with  the 
same  velocity,  and  in  the  same  direction ;  considered  horizontal- 
ly, therefore,  they  must  preserve  the  same  distance  from  each 
other. 

320.  In  the  foregoing  theory,  we  have  taken  it  for  granted  ; 
(1.)  that  the  force  of  gravity  is  the  same  throughout  the  whole 
range  of  the  projectile.  (2.)  That  it  acts  in  lines  parallel  to  each 
other.  (3.)  That  there  is  no  resisting  medium.  The  two  first 
suppositions,  although  not  strictly  conformable  to  fact,  are  attended 
with  no  material  error  in  practical  gunnery,  and  those  arts  to  which 
this  theory  is  subservient.  But  the  third  is  of  essential  importance 
to  the  truth  of  the  results  we  have  obtained.  We  can  readily  put 
the  theory  to  the  test  of  actual  experiment. 

The  initial  velocity  of  a  cannon-ball,  for  instance,  may  be 
obtained  with  considerable  accuracy,  by  either  of  the  following 
methods. 

321.  (1.)  Let  the  cannon  together  with  the  carriage  and 
other  weight  if  necessary,  be  suspended  like  a  pendulum,  so 
as  to  move   freely  in  the  direction  opposite  to  that  in   which  the 


Motion  of  Projectiles.  203 

ball  is  to  be  discharged.*  Upon  the  explosion  taking  place,  the 
cenlre  of  gravity  will  remain  unchanged,  that  is,  the  quantities  134. 
of  motion  in  opposite  directions  will  be  equal ;  consequently,  if 
the  motion  of  the  gun,  &c.,  be  made  so  slow  by  means  of  the 
attached  weight,  as  to  admit  of  its  velocity  being  taken  by  actual 
observation,  the  velocity  of  the  ball  will  be  as  much  greater  as  its 
mass  is  less.  Knowing  the  mass  of  each,  we  should  use  the  follow- 
ing proportion ;  as  the  mass  or  weight  of  the  ball  to  that  of  the  gun, 
carriage,  he,  so  is  the  velocity  of  the  latter  to  that  of  the  former. 

322.  (2.)  The  ball  may  be  discharged  into  a  large  block  of 
wood  suspended  so  as  to  move  freely  after  the  manner  of  a  pen- 
dulum,* and,  the  velocity  being  observed  as  before,  we  then  say 
as  the  mass  of  the  ball  to  that  of  the  pendulous  body,  so  is  the 
velocity  of  the  latter  to  that  of  the  fonner.  This  latter  method  is 
adapted  to  finding  the  velocity  at  different  distances  from  the  cannon. 

It  is  thus  found  that  the  velocity  of  a  cannon-ball  varies  ac- 
cording to  the  quantity  and  quality  of  the  powder,  the  size  of  the 
ball,  the  length  of  the  piece,  &ic.  At  the  commencement  of  the 
motion,  it  is  ordinarily  between  800  and  IGOO  feet  in  a  second. 

323.  With  a  velocity  equal  to  800  feet  in  a  second,  the   angle 

of  projection  being  45°,  for  instance,  the  horizontal  range,  greatest 306,307, 
elevation,  &-c.,  are  readily  determined    by  our  formulas. 

We  first  find  the  height  h  through  which  a  body  must  fall  to 
acquire  the  velocity  of  projection  800  feet,  and  double  this  height 
will  be  the  horizontal  range  required.  Now  to  acquire  a  veloci- 
ty of  800  feet  in   a  second,  a  body  must  fall  through  a  space  equal 

(800)2  800  ft.. ..log.. ..2,90309  277. 

64,4  ^ 

5,80618 
64,4.. ..log....  1,80389 

h  =2  9937,75         .         .         .         3,99729 


Range  =  19875,5  =  3,7  miles. 


*  It  will  be  seen  hereafter  at  what  point  in  the  pendulunr)  the 
impulse  must  be  applied  in  order  that  no  part  of  it  may  be  expended 
against  the  supports  from  which  the  pendulum  is  suspended. 


204  Dynamics. 

The   greatest    elevation    is    equal    to  h  multiplied    by  the  sine 
square  of  the  angle  of  projection,  that  is,  equal  to  h  (sin  45'^)^. 

h  =  9937,75ft.  log    3,99729 

45°  log  sin    9,84949 

9,84949 


Greatest  elevation  =  4969  feet  3,09627 

4969  wants  only  311  feet  of  a  mile. 


J- 


Moreover,     according     to    the     case    supposed,    we    have 
—  as  the  expression  for  t  the  time  of  flight. 

k  =  9937,75.. ..log....3,99729 
^  =  32,2  log   1,50786 


2)2,48943 
17'^57  1,24472 

9 


t  =  35,   14 

On  the  supposition  of  a  velocity  of  1600  feet  in  a  second,  the 
angle  of  projection  being  the  same,  we  should  have  for  the  hori- 
zontal range  79503  feet  or  15  miles>  for  the  greatest  elevation 
3,7  miles,  and  for  the  time  of  flight  3  minutes  and  38  seconds. 
So  great,  however,  is  the  resistance  of  the  air,  that  a  cannon-ball, 
under  the  most  favorable  circumstances,  is  seldom  known  to  have 
a  range  exceeding  3  miles;  the  path  described  is  not  strictly  a 
parabola  or  any  known  curve  ;  its  vertex  is  not  in  the  middle,  but 
more  remote  from  the  point  of  projection  than  from  the  other  ex- 
tremity ;  and  the  path  through  which  the  body  descends  is  less 
curved  than  that  through  which  it  ascends.  This  resistance  increas- 
es faster  than  the  velocity ;  so  that  in  the  slower  motions,  there  is 
a  nearer  approach  to  the  foregoing  theory,  than  in  those  which  are 
more  rapid,  as  is  apparent  to  the  eye  in  the  spouting  of  water,  and 
more  especially  of  mercury,  from  the  side  of  a  vessel.  To  treat 
of  this  resistance,  and  to  estimate  its  effects,  belongs  to  that  branch 
of  our  subject  which  has  for  its  object  the  motion  of  fluids  and  that 
of  bodies  immersed  in  them. 


Motion  down  inclined  Planes.  205 


.^^. 


Of  the  Motion  of  heavy  Bodies  down  inclined  Planes.  ^ 

324.  A  heavy  body  left  to  itself  upon  a  plane  surface  ^LiJZ,  Fig.  1 56. 
inclined  to  a  horizontal  surface  PIHN,  cannot  yield  entirely  to 
its  gravity.  A  part  of  the  force  derived  from  this  cause  is  em-  37. 
ployed  in  pressing  the  plane,  and  the  other  serves  to  bear  it  along 
the  plane.  It  is  necessary,  therefore,  to  decompose  its  gravity  into 
two  forces,  one  of  which  produces  the  pressure  upon  the  plane,  and 
the  other  the  motion  along  this  plane. 

32.5.  Let  G  be  the  centre  of  gravity  of  the  body  m,  or  the 
point  in  which  all  its  action  may  be  considered  as  united.  Let 
GB  be  the  space  through  which  it  would  fall  in  an  instant,  if  it 
were  free.  Let  GC  be  drawn  perdendicular  to  the  plane;  and 
suppose  a  plane  to  pass  through  GB,  GC,  this  plane  will  be  per- 
pendicular to  the  two  planes  KLHI,  IPNH,  since  it  passes  through  ^^°"'* 
the  straight  lines  perpendicular  to  these  planes.  If,  therefore,  we 
conceive  DE,  EF,  to  be  the  intersections  of  this  plane  with  KLHI, 
IPNH ;  DE,  EF  will  be  perpendicular  to  the  common  intersec-  Ceom. 
tion  i?J  of  these  two  planes.  3^^* 

Draw  GA  parallel  to  DE,  and  construct  the  parallelogram 
GABC  o{  which  GB  is  the  diagonal,  and  GA,  GC,  the  sides. 
We  may  suppose  that  gravity,  instead  of  urging  the  body  according 
to  GB,  urges  it  at  the  same  time  according  to  GC  with  the  veloci- 
ty GC,  and  according  to  GA  with  the  velocity  GA.  Now  it  is 
evident  that  G^C,  being  per|)endicular  to  the  plane,  cannot  but  be 
destroyed,  if  the  point  O  where  it  meets  the  plane  is  at  the  same 
time  a  point  common  to  the  plane  and  the  body  m. 

As  to  the  force  GA,  since  it  tends  neither  to  a[)proach  toward, 
nor  to  recede  from  the  plane,  it  cannot  but  have  its  full  effect. 
GA,  therefore,  represents  the  velocity  with  which  the  body  tends 
to  move,  and  with  which  it  would  move  in  the  first  instant. 

As  the  force  GA  is  in  the  plane  of  the  two  right  lines  GB, 
GC,  it  is  in  the  plane  DEF.  We  can  therefore  leave  out  of  con- 
sideration the  extent  of  the  two  planes  KLHI,  IPNH,  and  employ 
only  the  plane  D/Ji^  represented  in  figure  157,  so  that  the  body 
may  be  supposed  to  move  in  the  right  line  DE. 


206  Dynamics. 

326.  Since  the  force  G^  passes  through  the  centre  of  gravity 
G  of  the  body  m,  it  must  distribute  itself  equally  to  all  parts  of  ibis 

116.  body.  Tiierefore,  so  long  as  hiciion  is  supposed  to  have  no  influ- 
ence, the  body  can  have  no  motion  except  that  of  sliding  along  the 
plane,  that  is,  it  can  have  no  tendency  to  roll,  whatever  may  be 
its  figure,  provided  the  perpendicular  GB  meets  the  plane  in  a 
point  that  belongs  at  the  same  time  to  the  surface  of  the  body. 
This  would  not  be  the  case,  however,  as  we  have   seen,  if  the   per- 

196.  pendicular  did  not  meet  the  base  of  the  body,  or  the  surface  by 
which  it  rests  upon  the  plane.  The  influence  of  friction,  moreover, 
tends  to  produce  a  rolling  motion. 

327.  Since  the  body  m  must  describe  GA  in  the  same  time 
in  which  it  would  describe  GB  by  the  free  action  of  gravity,  if 
we  conceive  that  at  the  end  of  the  first  instant,  gravity  acts  anew ; 
as  it  communicates  in  equal  instants  equal  degrees  of  velocity,  by 
supposing  for  the  second  degree  of  velocity  communicated  in  a 
vertical  direction,  a  decomposition  similar  to  that  above  niade  for 
the  first  instant,  it  is  evident  that  the  second  parallelogram  will  be 
equal  in  all  respects  to  the  first.  We  accordingly  infer,  in  like 
manner,  that  the  force  perpendicular  to  the  plane  will  be  de- 
stroyed, and  the  force  parallel  to  the  plane,  and  equal  to  GA, 
will  be  added  to  GA.  By  reasoning  in  the  same  manner  for 
the  following  instants,  we  should  conclude  that  the  velocity  along 
the  inclined  plane  is  accelerated  by  equal  degrees ;  in  other  words, 
that  the  motion  of  heavy  bodies  down  an  inclined  plane  is  a  motion 
uniformly  accelerated.  Hence  all  that  has  been  said  upon  the  sub- 
ject of  motion  uniformly  accelerated,  is  strictly  applicable  to  the 
motion  that  takes  place  down  inclined  planes.  Consequently  in 
this  latter  case,  as  well  as  in  the  former,  the  velocities  are  as  the 
times,  the  spaces  described  are  as  the   squares  of  the  times,  or  as 

264,  &,c.  the  squares  of  the  acquired  velocities,  &c. 

328.  Therefore,  in  order  to  determine  the  motion  that  takes 
place  upon  a  plane  of  a  known  inclination,  we  have  only  to  find 
the  ratio  of  the  accelerating  force  to  gravity,  that  is,  the  ratio  of 
GA  to  GB.  Now  GA^  GB,  being  parallel  respectively  to  DE, 
DF,  the  angle  AGB  is  equal  to  EOF,  and,  the  angle  A  being  a 
right  angle  as  well  as  the  angle  F,  the  two  triangles  AGB,  EDF, 
are  similar ;  whence, 


273. 


Motion  down  inclined  Planes.  207 

DE  :  DF  :  :   GB  :   GA  ', 

that  is,  the  length  of  the  inclined  plane  is  to  its  height,  as  the  velocity 
which  gravity  commvnicates  to  a  free  body,  is  to  that  with  ivhich  it 
urges  the  body  along  the  inclined  plane. 

329.  Now  as  gravity  gives  to  a  free  body,  in  a  second  of  time, 
a  velocity  by  which  a  space  of  32,2  feet  are  described  uniformly 
in  a  second,  it  will  be  easy  to  determine  the  velocity  acquired 
by  a  body  in  the  first  second  of  its  descent  down  an  inclined 
plane.  If,  for  example,  the  length  of  the  plane  is  double  the 
height,  the  velocity  acquired  along  the  plane  during  the  first 
second,  will  be  half  of  32,2  feet ;  that  is,  at  the  end  of  the  first 
second,  if  gravity  ceased  to  act,  the  body  would  pass  over  1G,1 
feet  in  a  second. 

Having  thus  determined  the  velocity  for  the  first  second,  we 
shall  have  the  velocity  after  any  proposed  number  of  seconds, 
by  multiplying  this  by  the  number  of  seconds ;  also  the  space 
is  found  by  multiplying  this  first  velocity  by  half  the  square  of  267. 
the  number  of  seconds.  In  short,  it  would  be  easy  to  determine 
all  the  other  circumstances  of  the  motion  in  question,  by  articles 
267,  &,c.      We  hence  deduce  the  following  propositions. 

330.  If  two  heavy  bodies,  setting  out  at  the  same  time  from 
the  point  D,  descend,  one  along  the  plane  DE,  and  the  other  inFi<r.  153. 
the  direction  of  the  perpendicular  DF,  and  we  would  know  in 
what  part  of  the  plane  DE  the  first  would  be,  when  the  second 
had  arrived  at  any  given  point  A,  we  have  only  to  let  fall  from 
the  point  A  upon  DE  the  perpendicular  AB  ;  and  the  point  B 
will  be  the  place  sought.  Indeed  if  we  represent  by  g  the  velo- 
city that  gravity  communicates  to  a  free  body  in  one  second,  by 
calling  t  the  time  employed  in  describing  DA,  we  shall  have 

DA  =  \g  {^,  267. 

on   the  other  hand  the  velocity  acquired  in  a  second  by   the  body 

.    ^  X  DP 

that  descends  along  the  plane  DE,  is  - — j-r-=j — ;  accordinsjily    by 

calling  t'  the  time   employed   in  descending  from  D  to  B,  we  shall 
have 

2JJ}  —       ^^        X   jt      , 


208  Dynamics. 

whence, 

DA  :  DB  ::  1  g  t^  :  lJ<-PJ:  x  V'% 

::  DE   X    t""  :  DF  X   t'K 

But  by  similar  triangles,  ^ 

Goom.  DJ  :  DB  ::  DE  :  DF; 

213. 

consequently, 

DE  :  DF  :  :  DE  X  t^  •  DF  X  t'^; 

therefore, 

f^  =  t^     or     t'  =  t. 

Fif.lSQ.  331.  Hence,  if  DG  be  a  third  plane  described  by  a  third 
body  setting  out  from  D  at  the  same  time  with  the  other  two,  by 
drawing  frotn  the  point  .3  the  perpendicular  AC,  A,  B,  C,  are 
the  three  points  at  which  the  three  would  arrive  in  the  same 
time. 

332.  If  upon  DA  as  a  diameter,  we  describe  a  semicircum- 
ference,  it  will  pass  through   the  points  C  and  B,  since   the   angles 

Geom.  at  C  and  B  are  right  angles.  Consequently,  the  chords  DC, 
DB,  and  the  vertical  diameter  DA,  are  all  described  in  the  same 
time  ;  and  as  this  does  not  depend  upon  the  length  or  inclination 
of  the  chords,  we  may  draw  the  general  conclusion,  that  the  time 
employed  by  a  body  in  falling,  through  any  chord  of  a  circle,  drawn 
from  the  extremity  of  a  vertical  diameter,  is  the  same  as  that  employ- 
ed in  falling  through  this  vertical  diameter. 

333.  We   have   seen   that  g  being  the  velocity    communicated 

o-  X  DF  . 
to  a  free  body  in  one  second,  - — =r-p^ —  is  that  given  in  the  same 

time  to  a  body  that  descends  along  DE.  Let  t,  t',  be  the  times 
employed  in  describing  DF  and  DE  respectively  ;  we  shall 
have 

DF  =  igt\  DE  =  ^^  x\t''^; 


DE 


whence, 


Motion  down  inclined  Planes.  209 

which  gives,  by  multiplying  the  extremes  and  means  and  reduc- 


— 2 


or 


DF  X  t'^  =  DE  X  t\ 
or,  taking  the  square  root  of  each  member, 

DF  XI'  =  BE  X  t', 
in  other  words, 

i  :  i'::  DF  :  DE. 

In  like  manner,  if  t"  represent  the  time  employed  in  describing 
DG  ;  v/e  shall  have 

t  :  i"  :  :  DF  :  DG  ', 

whence, 

i'  :  i"  :  '.  DE  :  DG  ; 

that  is,  the  times  employed  in  describing  different  planes  of  the  same 
height,  are  to  each  other  as  the  lengths  of  these  planes. 

334.  The  velocity  of  the  body  which  descends  along    DF,  is 
^  /  at   the  expiration  of  the  time   i.     For  a  similar  reason,   the     267. 

velocity  of  the  body  that  descends  along  DE,  is  p —   X  i'  at 

the  expiration  of  the  time  f.  Accordingly,  if  we  call  v,  v,  the 
velocities  acquired  by  the  two  bodies  respectively  upon  arriving  at 
the  points  jP,  E,  we  shall  have. 


whence, 


u  :v   ::gt  :  ^-^  X  t', 


DF 

vgt=ugX-fT-^X  t\ 


DE 

But,  as  we  have  just  seen,  322. 

t  :  t'  :  :  DF  :  DE, 

which  gives 

Meek.  27 


210  Dynamics, 

,  _DFxi' . 
^  —      DE     ' 

substituting  for  i  lliis  value  in  the  above  equation,  we  shall  have, 

V  =z  u. 

Therefore,  if  several  bodies  descend  along  planes  differently  inclined, 
but  of  the  same  height,  they  will  have  the  same  velocity  upon  arriving 
at  the  same  horizontal  line. 


Of  Motion   along  curved  Surfaces. 

335.  If  a  body  without  gravity  and  without  elasticity,  de- 
scribes, in  virtue  of  a  primitive  impulse,  the  successive  sides  AB, 

Fig.l60.BC,  he,  of  any  polygon,  upon  meeting  each  side  it  will  lose  a  part 
of  its  velocity,  which  may  be  determined  in  the  following  man- 
ner. 

Let  us  suppose  that  the  body  moves  from  A  toward  B,  and 
that  when  it  is  at  B,  its  velocity  is  such  as  in  a  determinate  time, 
one  second  for  example,  would  cause  it  to  describe,  if  it  were 
free,  the  line  BF  in  AB  produced.  Having  erected  upon  BC 
from  the  point  B,  the  perpendicular  BE,  we  imagine  the  rectan- 
gular parallelogram  BDFE,  of  which  BF  is  the  diagonal,  and 
the  sides  of  which  are  in  the  direction  o{  BC  and  BE.  Instead 
of  the  velocity  BF,  we  may  suppose  that  the  body  has  at  the  same 
time  the  two  velocities  5i),  BE;  and  as  the  side  BC  prevents 
its  obeying  the  velocity  BE,  it  is  manifest  that  its  velocity  is  re- 
duced to  BD. 

If  from  the  point  B,  as  a  centre,  and  with  a  radius  BF,  we 
describe  the  arc  Fl,  Dl,  which  is  the  difference  between  BF  and 
BD,  will  accordingly  be  the  velocity  lost.  Now  DI  is  the  versed 
'^'^"  '  sine  of  the  arc  Fl,  or  of  the  angle  FBC,  made  by  the  two  contig- 
uous sides  AB,  BC.  Therefore  so  long  as  these  two  sides  make 
a  finite  angle,  the  body  will  lose  a  finite  part  of  its  velocity  upon 
meeting  each  of  the  sides. 

336.  But  if  the  angle  formed  by  the  two  sides  is  infinitely 
small,  the  velocity  lost  will  not  only  not  be   a  finite  quantity,   but 


Motion  along  curved  Surfaces.  211 

It  will  not  be  an  infinitely  small  quantity  of  the  first  order,  it  will 
only  be  an  infinitely  small  quantity  of  the  second  order.  In  es-  Cai.  4. 
tablishing  this,  the  question  reduces  itself  to  showing  that  the 
versed  sine  of  an  infinitely  small  arc  is  an  infinitely  small  quan- 
tity of  the  second  order  ;  and  this  may  be  done  thus.  CD  being  Fig-i6l- 
any  arc  of  a  circle,  and  BD  a  perpendicular  upon  the  diameter 
AC,  we  have  2^5."' 

AB  :  BD  :  :  BD  :  BC  ; 

hence,  if  CD,  (and  for  a  stronger  reason  BD)  be  infinitely  small, 
BC  the  versed  sine  of  CD,  will  be  infinitely  smaller  than  BD, 
since  it  is  contained  in  BD  as  many  times  as  BD  is  contained  in 
the  infinitely  greater  quantity  AB.  Therefore  BC  is  infinitely 
small  of  the  second  order. 

337.  Accordingly,  if  a  body  without  gravity  move  along  the  F'ig.162. 
curved  surface  ABC,  it  will  have  throughout  the  same  velocity. 

For  by  considering  this  curve  as  a  polygon  of  an  infinite  number  of 
sides,  since  the  sides  make  angles  infinitely  small  with  each  other, 
the  loss  of  velocity  at  the  meeting  of  each  two  adjacent  sides  is 
an  infinitely  small  quantity  of  the  second  order  with  respect  to 
the  original  velocity.  Consequently  the  sum  of  the  velocities 
lost  in  passing  over  an  infinite  number  of  sides,  that  is,  in  passing 
over  any  arc  ABC,  can  only  form  an  infinitely  small  quantity  of 
the  first  order.  Therefore  the  velocity  is  not  affected  by  this  cir- 
cumstance. Cal.  4. 

338.  We  come  now  to  the  motion  of  heavy  bodies  along 
curved  surfaces.  We  shall  consider  for  the  present  only  that 
which  takes  place  in  a  vertical  plane. 

339.  Accordingly,  let  AMB  be  a  section  of  a  curved  surface,  Fig.i63. 
made   by   a   vertical   plane,   and    the    path  described   by   a   body 
along  this  surface.     Let  us  consider   this  curve    as  a  polygon  of 

an  infinite  number  of  sides,  and  let  us  suppose  that  the  body  has 
just  described  the  small  side  LM.  As  its  meeting  with  the  side 
MN  cannot  occasion  any  loss  of  velocity  ;  it  will  describe  MN  336. 
with  the  velocity  which  it  had  in  M,  gravity  being  supposed  no 
longer  to  act  upon  it.  But  the  force  of  gravity  being  exerted 
according  to  the  vertical  MO,  urges  the  body  anew  as  it  would 
urge  one  upon  a  plane   of  the   same   inclination.     Consequently, 


212  Dynamics. 

if  we  imagine  the  velocity  MO,  which  gravity  tends  to  give  in 
an  instant,  decomposed  into  two  parts,  one  J\ID  perpendicular  to 
J\1N,  and  the  other  DO  or  ME  directed  according  to  MN;  we 
shall  see  that  it  is  by  virtue  of  this  last  that  the  velocity  of  the 
body  will  be  accelerated.  Now  by  letting  fall  the  perpendicular 
RN,  and  comparing  the  similar  triangles  MOE,  MNR,  we  shall 
have, 

NR  X  MO 


MN  :  NR  ::  MO  :  ME  = 


MN 


Let  us  suppose  that  the  different  points  of  the  curve  AB  are  re- 
ferred to  the  vertical  axis  BZ.     If  we  call 

BP,  X ;     PM,  y  ;     and  the  arc  BM,  s ; 

we  shall  have 

PQ  or  RN  =  —  ^  a? ;  and  MN  =  —  d  s. 

We  give  the  sign  —  to  these  quantities,  because  x  and  s  go  on 
diminishing  while  the  time  t  increases.  Let  g  be  the  velocity 
which  gravity  gives  to  a  free  body  in  a  second  ',  g  d  t  will  be  that 
267,  which  it  would  give  in  the  instant  d  i.  We  shall  therefore  have 
the  velocity  represented  by  AlO  as  follows,  namely,  MO  :=  g  d  t. 

Calling  V  the  velocity  which  the  body  has  when  it  arrives  at  M; 
d  V  will  denote  the  augmentation  received  during  the  time  dtj 
thus, 

ME  z=dv. 

Substituting  the  values  above  obtained  in  the  equation 

ME  =    ^^^Xil/Q 

MN 

we  shall  have, 

J  —  d  X  ,  ^  d  X  , 

^^=— d-sX5-^^  =  -^x  gdt, 

d  s 
But,  by  article  280,  d  s  z=  v  d  t,  ov  d  t  =  —,  or,  .s  being  consid- 

ered  as  decreasing  while  t  increases,  d  t  =  ;    whence,    by 

substitution, 


Motion  along  curved  Surfaces.  213 

,  d  X  —  d  s  ff  d  X 

d  s        ^  V  V 


V  d  V  z=z  —  g  d  X. 
The  integral  of  this  equation  is,  Cal.  83. 

^=C-gx, 

whence, 

v^  =:2  C  —  2g  X. 

In  order  to  deterntiine  the  constant  C,  let  us  suppose  that  the 
point  A  from  which  the  body  begins  to  fall,  is  elevated  above  a 
horizontal  line  passing  through  ^  by  a  quantity  BZ  =z  h.  It  is 
necessary,  therefore,  when  v  is  zero,  that  x  should  be  equal  to  h ; 
accordingly  we  have 

0  =  2  C  —  2gh, 

and  consequently 

2  C  =  2gh,0T  C  =zgh', 

whence,  by  substitution, 

v^  =  2gh  —  2  g  X  =  2  g  {h  —  x), 
=  2g  X  ZP. 

Now,  if  a  heavy  body  fall  through  the  space  ZP,  the  square  of 
the   velocity   which   it  will    have   upon   arriving   at    P,   will    be 

Therefore,  when  a  body  descends  along  any  curved  line,  it  has,  at 
any  point  whatever,  the  velocity  which  it  would  have  acquired  by 
falling  freely  through  a  space  of  the  same  perpendicular  elevation. 

Thus  the  velocity  which  a  body  successively  acquires  by  its 
gravity  in  descending  along  the  concavity  of  a  curved  line,  is  alto- 
gether independent  of  the  nature  of  this  curve. 

340.  Hence,  if  the  body,  after  having  arrived  at  the  lowest 
point  B  (the  tangent  to  which  I  supposed  to  be  horizontal),  meets 
the  concavity  of  the  same  or  of  any  other  curve,  touching  the  first 
in  B,  it  will  rise  upon  this  last  to  a  height  equal  to  that  from  which 
it  descended. 


277. 


214  Dynamics. 

Indeed,  let  us  suppose  that  the  body  is  actually  in    B,   or  that 
a;  =  0  ;  its  velocity  will  be  such,  that  we  shall  have 

v^  =  2  g  h  ,  or  u^  =1  2  g  h, 

by  calling  this  velocity  u  to  distinguish  it  from  the  other.  Let  us 
imagine  tiiat  with  this  velocity  it  ascends  along  any  curve  BM' ;  we 
shall  find  by  the  same  reasoning  as  that  above  pursued,  that  its 
velocity  in  any  point  M',  is  determined  by  the  equation, 

-dv'  =  ^,XgdU 

by  calling  v'  the  velocity  in  this  case,  and  5'  the  arc  BM',  and 
observing    that  v'  diminishes  according  as   t',  s',   and  x  increase 

280.    respectively.     Consequently,  putting  for  d  i  its  value  — ^,   we  shall 


have 


d  v'  ■=■  -^—^ —  01  v'  d  v'  =z  —  g  d  X  y 


V' 

and  by  integrating, 

D'2  z=  2  C—2gx. 

But,  when  a:  =:  0,  the  velocity  v'  is  w,  accordingly, 

v^  =  2  C—0, 
and  since 

u^  =  2  g  h, 

we  have 

v'^  =z  2  g  h  —  2  g  X. 

Now  when  the  body  ceases  to  ascend,  v'  =  0,  which  gives 

0  =  2gh  —  2gx; 

whence  we  deduce 

X  z=:  h. 

Therefore  the  point  at  which  the  body  will  have  arrived  in  any 
curve  BA',  will  be  at  the  same  height  as  the  point  A. 


Motion  of  Oscillation.  215 


341.  As  to  the  time  employed  in  describing  any  arc  AM  or 
3  of  the  curve  ;   since  d  t  : 
equal  v^2  gh  —  ^gx,  we  have, 


d  s 

AB  of  the  curve  ;    since   d  t  z=  ,     substituting    for    v     its 


dt=         -^^ 


s/2gh  —  2gx' 


280. 


so  that  it  would  be  necessary  by  means  of  the  equation  of  the 
curve  to  find  the  value  of  d  s  in  x  and  d  x,  and  having  substituted 
it  in  the  expression  for  d  t,  we  should  have  that  of  t  by  integrat- 
ing. 


Of  the  Motion  of  Oscillation. 

342.  We  have  seen  that  a  heavy  body  having  descended 
through  any  arc  of  a  curve  AB  must,  setting  aside  the  resistance 
of  the  air  and  friction,  ascend  again  to  the  same  height  in  a  curve 
BA'  which  has  at  the  point  B  the  same  horizontal  tangent  with 
BA.  Accordingly,  this  body  in  returning  would  describe  in  a 
contrary  direction  the  whole  extent  of  the  curve  A'BA ;  and  thus  340. 
would  continue  to  move  backward  and  forward  without  end. 
This  kind  of  motion  is  called  oscillation.  We  have  seen  what  is 
in  general  necessary  to  determine  the  duration  of  each  oscillation 
which  must  evidently  be  double  the  time  employed  in  describing 
the  arc  AB,  if  BA'  is  the  same  as  AB. 

When  the  curve  through  which  the  body  descends  is  circular, 
and  the  oscillations  take  place  through  small  arcs  only,  they 
have  this  remarkable  and  important  property,  that  their  duration 
is  not  sensibly  affected  by  the  extent  of  the  arc  AB  ;  so  that  the  Fig. 164. 
arc  AB  being  small,  as  four  or  five  degrees  only  at  the  most,  the 
body  will  always  arrive  at  B  in  the  same  time  very  nearly,  wheth- 
er it  set  out  from  the  point  A,  or  from  some  other  point  O,  taken 
between  A  and  B. 

Thus,  retaining  the  denominations  used  above,  and  designat- 
ing by  a  the  radius  JBCof  the  circle  BAD,  we  shall  have,  by 
the  nature  of  the  circle,  '^"S- 

'  iOl. 

i/  =  2  a  X  —  x^  or  y  =  \/2  a  x  —  x^ ; 


216  Dynamics. 

from  which  the  value  of  M  m,  or  d  s  or  \/rf  ^a  _|_  ^^  ^  is  readily 
found  to  be  as  follows,  namely, 

,  a  d  X 

Cal.  78.  d  S  = 


V/2  ax  —  x^ 


But  since  the  arc  BM  is  small,  x  is  small  with  respect  to  a,  and 
x^  may  be  neglected  when  taken  in  connexion  with  2  a  x  without 
material  error,  which  leaves 

,  adz 

a  s  =. 


•v/2  ax 


341.    Substituting  this  value  oi  d  s  in  the  expression  for  d  t,  we  shall 
have 

,.  —  adz  —  a  d  X 

d  t  :=. 


or,  since 


\/2  a  X  >/2  gh  —  ^  g  X         •v/4  \/a  Vx  Vg  ^/h  —  x 

—  ^  a  d  z 

>/a  \/"'  s/A  X  —  x^ 


a  ^/a  I  a 

—7=  =  %/«}  and  —7^  =:      — 


X  d  X 


~Sg 


Vl't 


a  d  X 

Now  as     /,-  o  expresses  the  element  of   an  arc  of  a 

V 2  ax  —  x''       * 

circle  of  which  the  diameter  is  2  a  and  the  abscissa  x  ;  so,  in  like 

^  h  d  X 
manner,    ~y,  — ^  expresses  the  element  of  an  arc  of  a  cir- 

cle whose  diameter  is  h  and  abscissa  x.  But  the  line  BZ  being 
h,  if  upon  BZ  as  a  diameter  we  describe  the  semicircle  BM'Z, 
M'  m!  will  be  this  element ;   so  that  we  shall  have 


Motion  of  Oscillation.  217 

whence 

^dx       _  djBM) 
Vh  X  —  z^  h 

Substituting  this  value  in  the  expression  for  d  t,  we  have 
and  by  integrating 


BM' 

X   —r-- 


We  have  therefore  only  to  determine  the  constant  quantity  C; 
and  it  will  be  seen,  that  when  t  =.  0,  that  is,  when  the  body  sets 
out  from  the  point  A,  the  arc  BM'  becomes  the  semicircumfer- 
ence  BM'Z ;  accordingly, 


\g 


BM'Z 

X  r— , 


whence 


therefore 


^  =  Jfx-f^: 


Ja^        BM'Z  U 

g^        h   '        <g    ^ 


BM 


=J 


a         ZM 


We  have  thus  an  expression  for  the  time  employed  in  describing 
any  arc  AM,  the  time  being  supposed  to  be  reckoned  in  seconds. 
But  when  the  arc  AM  becomes  AB,  that  is,  at  the  end  of  a  semi- 
oscillation,  the  arc  ZAP  becomes  ZM'B  ;  consequently,  by  calling 


\g 


,.  =  .|-x^. 


Mech.  28 


218  Dynamics. 

or,  

2  ZMB 


\g 


X         k 


Geom.     Now,  n  being  the  circumference  of  a  circle  whose  diameter  is  1, 
291. 


whence 


and  consequently 


or 


I   :  71  ::  h  :  2  ZMB  ; 
2  ZMB 

We  have  thus  an  expression  for  the  duration  of  an  entire  os- 
cillation ;  and  as  this  quantity  does  not  contain  h,  or  the  height 
from  which  the  body  falls,  and  which  determines  the  extent  of  the 
path  described  AB,  it  follows  that  the  time  f  does  not  sensibly 
depend  upon  the  extent  of  the  arc,  so  long  as  this  arc  is  very  small. 
Therefore,  the  oscillations  which  take  place  in  small  arcs  of  a  circle 
are  sensibly  isocronous  or  of  the  same  duration. 

343.  This  property  belongs  to  the  small  arcs  of  all  curves  in 
which  the  radius  of  the   evolute  at  the  lowest  point  is  not  zero  ; 
since  the  arcs  are  confounded  with  those  of  the  circle  by  which 
Cai.  79.  their  curvature  is  measured. 

If  we  would  know  the  error  liable  to  be  committed  by  taking 
this  value  of  f  for  the  duration  of  a  semioscillation  in  a  circle,  we 
proceed  thus ; 

Taking  the  value  found  above  for  d  s,  namely, 

7  a  d  X 

a  s  =z 


\/2  a  x  —  z^ 


CaJ.       we   reduce   it  to   a  series,  retaining  the   three  first   terms   only, 
'^      which  are    abundantly    sufficient  for  our  purpose,  and   we   shall 
have 


Motion  of  Oscillation.  219 

, ad_x_  /  a=    _L   3^^N 

by  substituting  this  value  in  the  expression  for  d  t,  namely, 


ds 


\/2gh  —  2gx 
and  reducing,  we  obtain 

^h  —  X 

As  we  know  already  the  integral  of  the  first  term,  we  shall  con- 
fine ourselves  to  finding  that  of  the  two  last.  Representing  it  by 
i  i",  we  shall  have 

\'a/^^dx    ,    Sx^dx\.y  \— ^. 

idt"  =  -l^f{-^--+-^-^){h-:c)      ^ 

To  obtain  the  integral  of  this  equation,  we  have  recourse  to  the 
method  laid  down  in  the  Calculus,  articles  128,  &:c.,  and  put  |  f 
equal  to  the  following  expression,  namely, 

I  t"  =  -  i  ^f  {-^  ^^  +  B  X  ^)  {h-  x)^ 

—  1  J-  Cfdxx~'^{h  —  x)~^+  D. 
The  co-efficients  are  then  determined  as  follows,  namely,  Cal.  129. 

64  a2'  ^  —        4  a        128  a^'  ^  —  8  a  "*"  256  a^' 

±  1 

Now  the  integral  of  x        ^  d  x  [h  —  x)       ^,  or  of 

dx  ,,1  hhdx 

or  ot  T-r  X 


s/hx  —  x'i  ^  h         \/hx  —  x2' 

•r-7  X  arc  BM  : 


220  Dynamics. 

therefore  the  whole  inicgral  is 


1  1 


IT        ^M'         /A     ,      9  A2  \    ,     -^ 

- ^7  X  -/r  X  (s^  +  256-^0  +  ^- 

To  determine  the  constant  D,  we  observe  that  when  x  =  k, 
t"  must  be  equal  to  0,  and  that  the  arc  BM'  becomes  BM'Z;  in 
this  case,  therefore,  we  have, 

Substituting  the  value  of  D  obtained  from  this  equation  in  the 
expression  for  i  f',  making  x  =^  h  in  order  to  have  the  entire  in- 
tegral, and  observing  that  BM'  becomes  then  zero,  the  result  will 
be 

\g  h       ^8  a  ^  256  a^J' 

But,  by  taking  n  for  the  ratio  of  the  circumference  of  a  circle  to 

•     J-  1         B3TZ        n  ,.     , 

Its  diameter,  we  liave  — , —  =  -^  ;  accordingly, 

\  g    \8  a    '    25b  a^/ 

Comparing  this  value  of  t''  with  that  of  f  found  above,  we  shall 
have. 


t'  :  f 


or 

~      V8  «        256  a^J' 

a  quantity  in  which  -  is  the  versed  sine  of  the  arc  described  during 
a  semioscillation,  radius  being  1. 


Pendulums.  22 1 

Suppose  i'  equal  to  1",  and  that  the  arcs  described  on   each 
side  of  the  vertical  are  5°.     The  versed  sine  of  5°  is  0,0038053, 

radius  being  1.     Consequently,  k—  =  0,0004757.     With  respect 

to  the  term  ^^^r-. — 5,  it  is  less  than  a  unit  of  the  sixth  place. 
The  error  in  each  oscillation  will,  therefore,  be 

t"  =1  \"  X  0,0004757  =  0'^0004757. 

Thus,  if  a  body  descend  by  the  action  of  gravity  along  a  circu- 
lar curve,  and  describe  arcs  infinitely  small  on  each  side  of  the 
lowest  point  in  a  second  of  time,  the  duration  of  each  oscillation, 
no  allowance  being  made  for  friction  or  the  resistance  of  the  air, 
would  differ  only  0",0004757  from  that  of  an  oscillation  through 
an  arc  of  5°  on  each  side  of  the  lowest  point,  so  that  in  a  day  or 
during  24  X  CO  X  60  =:  86400'^  vibrations,  the  difference  would 
amount  to  86400  X  0'',0004757  or  41".  Thus  a  pendulum  of 
the  length  required  to  vibrate  seconds,  and  performing  its  oscilla- 
tions through  arcs  of  5°  on  each  side  of  a  vertical,  would  lose 
only  4V  a  day,  when  compared  with  one  vibrating  in  arcs  infinitely 
small. 

If  the  arcs  described  on  each  side  of  the  vertical  were  only  1°, 
the  versed  sine  of  which  is  0,0001523,  the  daily  loss  would  be 
only  1",64,  that  is,  1|  nearly,  and  for  half  a  degree,  the  loss 
would  be  0^,41  or  |  of  a  second  daily. 


U^J 


Of  Pendulums. 

344.  What  we  have  said  is  particularly  applicable  to  pendulums.  Fig.l65. 
By  a  pendulum,  is  to  be  understood  a  rod  or  thread  suspend- 
ed at  one  extremity  from  a  fixed  point,  and  supporting  at 
the  other  extremity  one  or  several  bodies.  It  is  called  a  simple 
pendulum  when  it  is  supposed  to  consist  merely  of  a  mass  or 
weight  sustained  by  a  thread  or  rod  without  gravity,  and  when  at 
the  same  time  this  mass  is  of  a  diameter  very  small  relative  to 
the  length  of  the  pendulum.  We  shall  speak  for  the  present  only 
of  the  simple  pendulum. 


223  Dynamics. 

When  the  pendulum  CB  is  drawn  from  its  vertical  position,  the 
force  of  gravity  acting  according  to  the  vertical  line  AM  is  not 
wholly  employed  in  moving  the  body ;  a  part  is  exerted  against 
the  point  C.  Let  therefore  the  whole  force  of  gravity,  represented 
by  AM,  be  decomposed  into  two  others,  represented  the  one  by 
AN,  directed  according  to  CAN,  which  will  be  destroyed,  and 
the  other  by  AP  which  urges  the  body  along  the  arc  AB.  Now 
as  the  radius  CA  is  perpendicular  to  the  arc,  it  will  be  seen  that  the 
motion  is  here  decomposed  in  the  same  manner  as  in  the  case 
above  considered,  where  the  body  is  supposed,  without  any 
material  connexion  with  C,  to  descend  along  the  arc  AB,  which 
has  for  its  radius  the  length  AC  of  the  pendulum.  Accordingly 
every  thing  which  we  have  said  is  applicable  to  pendulums.  The 
following  are  some  of  the  consequences  which  are  derived  from 
the  preceding  investigation. 

345.  We  have  found  for  the  duration  t  of  an  oscillation,  the 
following  expression,  namely, 

g 
Hence,  for  another  pendulum  whose   length  is  a\  and  which  is 
urged  by  a  different  force  of  gravity,  or  one  that  is  capable  of 
giving  the  velocity  g'  in  a  second,  we   shall   have,  by  calling  V  the 
duration  of  an  oscillation  in  this  second  case, 

i'  =  n  V^I; 
g 

hence  we  derive  the  proportion,  ' 

y  .  //  •  .       A./  a     .  „  A./  a'    .  .    A./  «     •  A./  «'    .  •    ^  "■   .  "^  "■' 

t  :  v   :  '.  It  -y —  :  n  \ —  :  :    -V •  \  —  :  .       —  :  ■    — -, 

g  g'  g  g'  V  8      V  s' 

that  is,  if  two  pendulums  of  different  lengths  are  urged  by  different 
gravities,  the  durations  of  the  oscillations  are  as  the  square  roots  of 
the  lengths  of  the  pendulums,  divided  by  the  square  roots  of  the 
quantities  tvhich  denote  these  gravities. 

346.  As  gravity  is  the  same  in  the  same  place,  we  shall  have 
for  pendulums  of  different  lengths  vibrating  in  the  same  place  or 
same  part  of  the  earth,  g^  ^=^  g,  and  consequently  in  this  case  the 
proportion  becomes 


Pendulums.  223 

t  :  t'  :  :  \/  a     :  ^  a'    ', 

that  is,  in  the  same  'place  the  durations  of  the  oscillations  are  as  the 
square  roots  of  the  lengths  of  the  pendulums. 

347.  But  if  the  same  pendulum  be  successively  exposed  to 
the  action  of  two  different  gravities,  a  being  equal  to  a',  the  pro- 
portion 


t  :  V 


:  :   \f_^    :    ^.^ 


becomes 


g  g' 


Vg-'  :  v/g-j 


in  other  words,  the  durations  of  the  oscillations  of  the  same  pendu- 
lum in  different  places  are  inversely  as  the  square  roots  of  gravity. 

348.  Let  n  be  the  number  of  oscillations  or  vibrations  made 
by  the  pendulum  a  in  a  given  time,  as  one  hour  or  3600",  we 

3600" 

shall   have   t  =.  — .     For  the  same  reason,  if  we  represent 

by  n'  the  number  of  vibrations  made  by  the  pendulum  a'  in  the 

3600" 

same  time,  we  shall  have  t'  =  — — ■ —  ; 

n' 

accordingly, 

_  3600" 
n 

that  is,  the  numbers  of  vibrations  made  in  the  same  time  by  two 
pendulums  of  different  lengths  are  inversely  as  the  durations  of 
their  respective  vibrations.     Consequently,  since 


t:t".  :  V-^   :   V^, 


g  g' 


n  :  n'  ::  \] ^L   :   V-^, 

g'  g 

that  is,  the  number  of  vibrations  made  in  the  same  time  by  two  pen- 
dulums of  different  lengths,  and  which  are  urged  by  different  gravi- 
ties, arc  in  the  inverse  ratio  of  the  square  roots  of  the  lengths  of  the 


224  Dynamics. 

pendulums  divided  by  the  square  roots  of  the  gravities ;  so  that  if 
the  gravities  are  the  same,  the  number  of  vibrations  will  be  recip- 
rocally as  the  square  roots  of  the  lengths  of  the  pendulums ;  and 
if  the  lengths  are  the  same,  the  number  of  vibrations  will  be  direct- 
ly as  the  square  roots  of  the  gravities. 

349.  Hence  if  the  same  pendulum,  carried  to  different  parts 
of  the  earth,  does  not  make  the  same  number  of  vibrations  in  the 
same  interval  of  time,  it  is  to  be  inferred  that  gravity  is  not  the 
same  in  these  places,  and  the  number  of  vibrations  actually  made 
in  the  same  time  by  the  same  pendulum  in  two  different  places, 
will  furnish  the  means  of  ascertaining  the  relative  intensities  of 
gravity  at  these  places.  It  is  by  experiments  of  this  kind,  taken 
in  connection  with  the  foregoing  proposition,  that  we  are  now  as- 
sured of  the  diminution  of  gravity  as  we  approach  toward  the  equa- 
tor, and  on  the  other  hand  of  its  augmentation  as  we  proceed  from 
the  equator  toward  either  pole. 

350.  If  we  call  t  the  time  employed  by  a  heavy  body,  falling 
freely,  in  describing  the  diameter  BD  or  2  a,  we  shall  have 

o  gt^~        a         t^ 

273.  2  a  =  ^  or  -  =  -7-  J 

2  g        4  ' 

Fig.  164.  whence  c,    ^    %-  ~7\^  h 

Ajl  =  ^t.      ' 
g 

Substituting  this  value  in  the  equation, 

t'  =  n  \]Z, 
g 


we  obtain 


which  gives 


t'  =  \nt    or    \t'  =  \nt. 


It'  :  t 


that  is,  the  duration  of  the  descent  through  any  small  arc  AB  is 

to  the  time  of  falling  through  the  diameter,  as  the   fourth  of  the 

Geom.    circumference  of  a  circle  is  to  its  diameter.     But  the  fourth  of  the 

292.       circumference   is   less   than  the  diameter;    consequently   a   body 


•         Line  of  sioifiest  Descent.  225 

employs  less  time  in  descending  along  a  small  arc  of  a  circle 
of  which  the  inferior  tangent  is  horizontal,  than  it  would  employ 
in  falling  through  the  diameter ;  and  since  the  time  required 
to  pass  through  the  diameter  is  the  same  with  that  required  to 
describe  any  chord  AB,  it  will  be  seen,  that  a  body  would  pass  332, 
sooner  from  A  to  B,  by  descending  along  the  arc  AB,  than  by 
moving  through  the  straight  line  AB.  Therefore,  although  the 
straight  line  is  indeed  the  shortest  way  from  one  point  to  another, 
it  is  not  that  which  requires  the  least  time  for  the  passage  of  a 
heavy  body. 

Of  the  Line  of  swiftest  Descent.  ///fj44/^ 

351.  Not  only  is  not  a  straight  line  that  along  which  a  heavy 
body  would  proceed  in  the  shortest  time  from  one  point  to  another, 
out  of  the  same  vertical,  but  it  is  not  the  arc  of  a  circle  w'hich 
answers  to  this  description  ;  it  is  the  arc  of  another  curve  which 
may  be  found  in  the  following  manner. 

Suppose  ./JJ^ii?  to  be  the  curve  sought,  or  that  through  which  Fig.i66. 
a  heavy  body  would  pass  in  the  least  time  from  a  given  point  A^ 
to  a  given  point  B.  If  we  take  in  this  curve  two  points  M,  m/, 
infinitely  near  to  each  other,  the  arc  M  m'  must  also  be  described 
in  less  time  than  any  other  arc  passing  through  these  same  points 
M,  m',  since  these  two  points  may  be  taken  as  the  very  points 
in  question.  Having  taken  the  point  N  infinitely  nearer  to  Mm/ 
than  M  is  to  m',  suppose  infinitely  small  straight  lines  M  N,  Nm'  to 
be  drawn  ;  since  the  lime  of  describing  M  m  ml  must  be  a  mini- 
mum, it  follows  that  the  difference  between  the  time  of  passing 
through  M  m  m'  and  the  time  through  MN  m',  which  is  the  differ- 
ential of  the  time,  must  be  zero. 

Through  the  points  M,  N,  m\  draw  the  horizontal  lines  MB^ 
m  P\  m'  P",  and  through  A,  the  vertical  line  AC.  Call  AP,  x  ; 
PM,  y ;  AM,  s ;  and  suppose  M  m  =  m  m',  or  that,  c?  5  is  con- 
stant. Then  m  r  =:  d  x,  r  M  =z  d  y,  m  r'  =  d  x  -\-  d  d  x, 
r'  m'  ^=  dy  -\-  d  d  y.  Let  u  be  the  velocity  with  which  the  body 
describes  M  m',\i  will  be  the  velocity  with  which  MN'xs  described  ; 
and  u  -\-  du  will  be  that  with  which  m  m'  and  Nm'  will  be  described. 
Mech.  29 


226  Dynamics. 

Therel'orc  the   time  ol  passins;   through   Al  m  will   be  — ,  and  the 

24.  " 

•111  ^  ^ 

lime  through  m  in'  will  be  — j— -r"- 
^  u-\-  du 

From  the  points  M  and  m'  as  centres,  and  with  the  radii  JfJV, 
mm',  describe  the  arcs  JVn,  mi;  then  comparing  the  triangles 
N  m  n,  JV  m  t,  with  the  triangles  M  m  r,  m  m'  r',  we  shall  have 

n  m  =.  JS  m  X  -r^, 
a  s 


and 


Whence 


a  s 


MJV=ds  —  JVm  X   4^, 
a  A- 


JV m'  =  d  s  4-  JVm  X  — ^^ -■ 


Therefore  the  time  through  J[1JV  will  be 

ds  —  iVra  X   -T^ 
a  s 


and  the  time  through  JV  m'  will  be, 


ds^Nm^^-y-±^ 


u  -\-  du 
We  have,  therefore, 

as  d  s  d . 


d  s 


'  u  -\-  du  u        u  -\-  du         ' 


an  equation  which  reduces  itself  to 

d  s   \    u  -j-  d  u  u  /  '  \"/ 


Ldne  of  smftest  Descent.  227 

then,  by  integrating,  we  have 

dy        d  s         ^  J  J 

— ^  =  -^,  or  L  d  y  z=z  u  d  s. 

But  since  the  velocity  u  is  equal  to  that  which  the  body  would 
acquire  in  falling  from  the  height  AP,  we  have 

u^  =  2  g  X.  277. 

Therefore, 

C  d  y  =  d  s  \/  2  g  X, 
and 

Cdy^  =  2gxds^  =  2g.x{dx''  ^dy"")', 

whence  we  deduce 

d  X  ^2Yx 


d  y  = 


^C'  —  2gx 


To  determine  the  constant  C,  it  will  be  observed  that  when 
\/  2  g  X  =  C,  we  have  d  y  =z  d  s,  answering  to  the  lowest  point  R, 
of  the  curve,  where  d  xz=  0,  and  x  :=  h;  therefore  if  we  call  v  the 
velocity  which  the  body  will  have  at  the  point  where  \^2g  x  =  C, 
the  equation  C  d  y  =z  u  d  s  becomes  C  d  s  z=  vd  s,  which  gives 
C  =zv.  And  if  we  call  h  the  corresponding  height  AC,  we  shall 
have 

v^  =  2gh; 
hence 

C^  =  2gh. 
Therefore 

,     d  X  \/2g  X   d  X  s/~x 

^         \/-2gh  —  2gx  \/h—  X 

is  the  equation  of  the  curve.     But  the  belter  to  understand   this 
curve,  let  us  give  another  form  to  the  equation. 

Imagine  the  vertical  line  TIT)  drawn  through  the  point  72, 
where  d  y  :=■  d  s  ;  and  having  produced  PM  to  O,  call  AD,  a ; 
OR,  x' ;  and  OM,  y'.  Then  x  =  h  —  x',  y  =  a  —  /, 
d  X  =:  —  d  x',  d  y  =  —  d  y' ;  substituting  these  values,  we  have 

d  x'  Vh  —  z'         h  d  X  —  x'  d  x 


dy'  = 


•s/xi  y/  h  x'  ~  xi  2 


\  h  d  x'  —  x'  d  x'  i  h  d  x' 


277 


228  Dynamics. 

Therefore 

Imagine  that  upon  DR  or  h,  as    a   diameter,  is  described  the 
semicircle  DER.     We  shall  have  OE  =  A^hxi'ZZ'^j^  and  ihe  arc 

RE=  rAAl^. 

t/ 


Cal.117.  t/     VAz'  — z'2 

we  have  therefore  generally, 

OM—  O  +  OE  +  RE. 

To  determine  the  constant    C',  it  must  be  observed  that  when 
x'  =  0,  we  have  y'  =:  0.     Therefore,  since  OE  and  RE  then  be- 
come zero,  we  have    C  =  0;  consequently   OM  =i  OE  -}-  RE ; 
and    the   curve   sought  is   therefore    the   common  semicycloid,  of 
Cal.  36.  which  DER  is  the  generating  circle,  and  AD  the  semibase. 

The  only  thing  which  remains  to  be  determined,  is  the  quan- 
tity h ;  for  the  only  things  given  are  the  two  points  A  and  B, 
through  which  the  body  is  to  pass.     A  is  determined  in  this  manner. 

Having  drawn  the  vertical  BK,  which  meets  in  K  the  hori- 
zontal line  AK  passing  through  the  point  A,  we  describe  upon 
AK  as  a  semibase,  the  semicycloid  AVT,  that  is,  a  semicycloid 
of  which  the  generating  circle  has  AK  for  the  length  of  its  semi- 
circumference.  And  having  drawn  AB  cutting  this  cycloid  in  V^ 
we  draw  VK,  and  parallel  to  VK,  through  the  point  B,  we  draw 
BD,  which  determines  AD  for  the  semibase  of  the  cycloid  sought, 
that  is,  for  the  semicircumference  of  its  generating  circle.  This 
construction  is  founded  upon  the  circumstance,  that  the  cycloids 
AVT,  ABR,  which  have  their  bases  upon  AD,  and  the  point  A 
common,  are  similar,  as  may  be  easily  shown.* 

*  Since  the  diameters  of  circles  are  as  their  circumferences,  or 
as  their  semicircumferences  ; 

DR  :  KT  :.  DA  :  KA; 


Georn. 
287. 


but 

DR  :  KT  :  :  DB  :  KV, 
hence 

DB  :  KV  :  :  DA  :  KA. 


Lijie  of  swiftest  Descent.  229 

352.  We  have  supposed  the  body  to  have  no  velocity  on  its 
leaving  the  point  A.  But  if  it  had  already  acquired  a  certain 
velocity  in  a  given  direction,  the  origin  of  the  curve  would  be 
at  some  higher  point.     The  equation  C  d  y  =  u  d  s,  found  above, 

gives  -^  =  ^  ;  whence  the  constant   C  must  be   such  that,  the 

initial  velocity  being  divided  by  it,  the  quotient  will  be  equal  to 
the  sine  of  ihe  angle  made  by  the  direction  of  this  initial  velocity 
with  the  vertical,  a  condition,  which,  with  the  other,  that  the  body 
must  pass  through  A  and  B,  will  determine  the  cycloid  for  the 
case  in  question. 

353.  Besides  this  property  of  being  the  curve  of  swiftest 
descent  in  an  unresisting  medium,  the  cycloid  is  on  several  otiier 
accounts  quite  remarkable.  It  has,  for  example,  this  singular 
property,  that  whatever  be  the  point  X,  from  which  a  body  be- 
gins to  descend  along  the  concave  part  of  the  curve,  it  arrives 
always  at  the  lowest  point  R  in  the  same  lime.  This  property  is 
thus  proved. 

Calling  t  the   time,   and  s  the   arc  RM  corresponding  to  any 

point  M,  where  the  body  is  found  at  the  end  of  the  time  t,  we 

d  s 
h^ve  d  t  = .      Now   designating   by   A'   the   height   of   X 

above  the  horizontal  line  OM,  we  have  u  =  \/2  g  {h'  —  x').     More-    211. 
over,  it  is  easy  to  infer  from  the  value   of  d  y',  found  above,   that 


d  s  -=2 
hence 

d  t  =. 


d  x'  y/h 


therefore. 


dx>  ^h  \  h  1  Ih'  dz' 

2 g  {h'  x'  —  x'^)  \'Z  g         ^  h'         ^h'  x'—x/*' 


t  = 


h'  \2g         J     ^//^'z'  — x'^ 


whence,  reasoning  as  above,  and  calling  i'  the  whole  time  employed     339. 
in  falling  from  X  to  R,  we  conclude  that 


230  Dynamics. 

^   —  h'  ^  SJ2g  ^     2  ' 

TT  being  the  ratio  of  the  circumference  to  the  dianneter.     There- 
fore 


t' 


">|2i' 


that  is,  the  time  t'  is  independent  of  the  height  h'  from  which  the 
body  sets  out. 


Of  the  Moment  of  Inertia. 

Fig.  167.  354.  Let  M,  m',  m",  be  any  masses  without  gravity,  and  let 
them  be  considered  as  points  situated  in  the  same  plane  with  the 
point  F,  and  connected  together,  and  with  the  point  F,  in  such  a 
manner  as  not  to  admit  of  any  change  in  their  relative  distances, 
or  of  any  motion  except  about  the  point  F,  or  nbout  an  axis  pass- 
ing through  F,  perpendicularly  to  tlie  plane  in  which  they  are  sit- 
uated. Let  us  suppose  that  these  masses  receive  at  the  same  time 
impulses  according  to  the  lines  w,  w',  w",  directed  in  the  above 
plane,  and  such,  that  if  the  masses  were  free,  they  would  have  ve- 
locities represented  by  these  lines  respectively,  it  is  proposed  to  de- 
termine the  motion  that  would  ensue. 

We   decompose,  according  to  the  principle  of  D'Alembert,  the 
133.    velocities  w,  w',  w",  each   into   two   others,  one  of  which  shall  be 
effective,  and  the  others  such,  that  if  the  masses  m,  m',  m'',  had  re- 
spectively only  this  velocity,  they  would  remain  in  equilibrium. 

Now  it  is  manifest  that  the  velocities,  which  the  bodies  are 
supposed  to  have,  since  they  admit  only  of  a  rotation  about  the 
point  JP,  must  be  perpendicular  to  the  radii  r,  r',  r'^  Moreover, 
in  order  that  these  velocities  may  take  place,  that  is,  not  mutually 
disturb  each  other,  it  is  necessary  that  they  should  be  proportion- 
al to  these  radii,  or  to  the  distances  respectively  from  F.  Accord- 
ingly, the  communicated  velocities  tw,  xo'^  w",  being  decomposed 
into  the  effective  velocities  v,  v',  v",  and  the  velocities  u,  u',  u", 
with  which  the  masses  would  be  in  equilibrium  about  the  point  F, 


Moment  of  Inertia.  231 

we  have 

V  :  v'  :  :  B.  :  r',  and  v  :  i/'  :  :  b.  :  n"  (i). 

Letting  fall  from   F  the  perpendiculars  a,  a',  a",  upon  the  direc-     63. 
lions  of  the  velocities  w,  u',  u"^  we  obtain, 

M  •  M  •  a  -j-  m!  '  u'  •  a'  —  m''  •  u"  '  a"  =  0. 

Now  if  we  let  fall  also  the  perpendiculars  c,  c',  c",  upon  the  direc- 
tions w^  w',  w",  we  shall  have,  by  article  62, 

M  •  M  •  a  +  M  •  v  •  R  =  M  •  w  '  c, 
or 

M  •  U   •  a    =Z    M  •  W  '  C   M'V'R. 

In  like  manner, 

m'  •  w'  •  a'    z=    m'  •  ■u/  •  c'  —  m'  '  v'  •  r'. 

If  from  the  sum  of  the  two  first  of  these  tiiree  equations  we  sub- 
tract the  last,  we  shall  have 

n  '  u  '  a  -\-  u'  '  u'  •  a'  —  u"  .u"  •  a", 

or  0,  equal  to  the  expression  below,  thus, 

0    =     M  '  10  '  c    -{-    m'  •  w'  '  c^     —     m''  •  w'"'  •  c'' 
—  n  •  V  '  R    —    m'  •  v'  '  r'    —    m"  •  v'^  '  r". 

But  the  above  proportions  (i)  give  v'  = and  v'^  = ; 

substituting    these   values   for   v'   and   v^',   the   equation   becomes 
0    =  M  '  to  '  c    -\-  m'  '  w'  '  c'  —  m'^  •  iv'^  •  d' 
m'  •  w  •  r'2         m"  •  V  •  r"2 

M  •  V  •  R — 

R  R 

=    ^\  •  w  '  c  •  ■\-  ^\'  '  xo'  •  c'  —  m''  •  w"  '  d' 

(m  •  Rg  -f-  M^  •  R^2  _j_  m''  •  r"2) 


whence 

M  •  ?«  •  r  -f-  v\'  •  w  '  d  —  m"  •  to"  '  c' 


M  •  R^  4-  m'  •  r'^  -f  M'"  •  U"5 


-     X     R. 


232  Dynamics. 

Now  the  numerator  of  this  fraction,  since  it  expresses  the  sum 
of  the  moments  of  the  forces  tn  •  w  '  c,  he,  is  equal  to  the  moment 
of  their  resultant.  If,  therefore,  we  call  this  resultant  q,  and 
its  distance  from  the  point  F,  D ;  the  sum  of  the  moments  will  he 
Q  X  D'  Moreover  the  denominator  of  the  above  fraction,  being 
the  sum  of  the  products  of  each  mass  or  particle  into  the  square 
of  its  distance  from  JP,  if  we  represent  in  general  any  one  what- 
ever of  these  masses  by  m,  and  its  distance  from  F  by  r,  the  sum 
of  these  products  may  be  represented  by  the  abridged  expres- 
sion r  m  r^,  {  r  denoting  the  word  sum)  ;  we  have  accordingly  for 
the  velocity  of  any  given  point  m,  whose  distance  from  the  axis 
JPis  FM  or  R,  the  following  expression, 

also, 


X   b: 


^  X  D  =  ~  rmr^. 

355.  Although  we  have  supposed  that  all  the  forces,  and  flll 
parts  of  the  system  are  in  the  same  plane,  it  will  be  perceived 
that  we  should  arrive  at  the  same  result,  if  they  were  in  planes 
parallel  to  each  other,  and  perpendicular  to  the  axis  of  rotation, 
provided  that  all  parts  of  the  system  admit  only  of  a  rotation 
about  a  fixed  axis. 

356.  Accordingly,  as  a  solid  body  of  whatever  figure  may  be 
considered  as  an  assemblage  of  material  points,  thus  connected 
together,  we  may  say  generally,  that  when  a  body  L  of  ivhatever  Jig- 

F'ig.l6S.  ure,  and  urged  by  forces  of  whatever  number  and  magnitude,  can 
have  no  other  motion,  except  a  motion  of  rotation  about  a  fixed  axis 
AB,  situated  within  or  without  the  body,  the  velocity  belonging  to 
any  given  point  is  found  by  taking  the  sum  of  the  moments  of  all 
the  forces  {or  the  moment  of  the  resultant),  dividing  this  sum  by  the 
sum  of  the  products  of  the  several  parts  of  the  body  into  the  squares 
of  their  distances  respectively  from  the  axis  of  rotation,  and  mul- 
tiplying the  quotient  by  the  distance  of  the  point  in  question  from 
this  same  axis. 

Fig.  169.  357.  Let  G  be  the  centre  of  gravity  of  the  body  L,  and  let 
us  suppose  that  while  any  point  m,  in  turning,  describes  during 
an  instant,   the   infinitely   small   arc   v,   the   centre  of  gravity  G 


Moment  of  Inertia.  233 

would  describe  the  arc  GG'  perpendicular  to  FG ',  through  the 
point  G'  let  the  line  G'K  be  drawn  parallel  and  equal  to  GF. 
Instead  of  supposing  the  body  to  turn  about  JP,  we  may  imagine  it 
carried  parallel  to  itself  with  a  velocity  equal  to  GG',  and  that  at 
the  same  time  its  several  parts  turn  about  a  movable  point  G 
with  such  a  velocity  that  by  taking  G'K  =.  GF,  the  point  K 
would  describe  the  arc  KF,  equal  to  G'G)  for,  on  this  supposition, 
the  point  J"  of  the  body  L  would  still  remain  stationary.  Now  the 
body  in  this  case  being  free,  the  resultant  of  all  the  motions  of  rota- 
tion about  the  movable  point  G  is  zero.  Consequently  the  result-  79. 
ant  of  all  the  motions  with  which  the  body  is  actually  urged  is  no 
other  than  that  which  the  body  L  would  have,  impressed  with  the 
velocity  G  G' ;  that  is,  this  force  must  be  perpendicular  to  FG,  and 
equal  to 

L  X  GG, 

the  mass  of  the  body  being  denoted  by  L.     Now  since  the  parts  of 
the  body  describe  similar  arcs,  we  have 

FM  :  FG  ::  V  :   GG'  =    ^^^^: 

FM    ' 

therefore,  the   resulting  force  of  all  the  motions  of  rotation  about 
the  point  F,  is 

Lx  FGxv 
FM 

But  although  this  resultant  is  the  same  as  if,  the  body  being 
free,  the  centre  of  gravity  had  received  the  velocity  GG^  still  it 
will  be  seen  that  this  resultant  does  not  pass  through  G,  but 
through  some  point  O  of  FG  produced  ;  since,  the  more  remote 
parts  having  the  greater  force,  the  resultant,  while  it  falls  on  the 
same  side  of  F  with  the  centre  of  gravity,  must  pass  at  a  greater 
distance  from  F  than  this  centre.  Designating  this  distance  FO 
at  which  the  resultant  passes  by  D',  we  shall  have  for  the  moment 
of  the  resultant 


FM 


If  now,  at  the  instant  when  the  forces  m  •  w  •  c,  &.c.,  above  con-     354. 
sidered,  begin  to  act  upon  the   parts  of  the  body,  there  be  opposed 
to  them,  at  the  distance  J)',  a  force  equal  to  that  just  determined; 
Mech.  30 


234  Dynamics. 

that  is,  equal  to  the  whole  effort  which  the  above  mentioned  forces 
would  exert  upon  the  body,  there  would  evidently  be  an  equilibri- 
um ;  but  in  this  case  the  moment  — ^^-jy X    X>'    must   be 

equal  to  the  moment  q  X  D',  accordingly,  since 


X  ^  =  /i/T'/  '"  ^' 


we  shall  have 

LX  FGxv 


FM 


^  ^'  =  -mJ'"'"^' 


and,  consequently, 


_  j^  Xfmr^  X    FM  _   X 


FMxLxFGxv  ~  L  X  FG' 

358.  We  hence  derive  the  general  conclusion,  that,  if  any 
number  whatever  of  forces,  directed  in  any  maimer  we  please,  in 
'planes  perpendicular  to  the  axis  of  rotation,  act  upon  a  body,  and 
are  capable  of  producing  only  a  ^notion  about  this  axis ;  (1.)  The 
force  thus  exerted,  will  be  equal  to  the  mass  of  the  body  multiplied 
by  the  velocity  belonging  to  the  centre  of  gravity  ;  which  velocity  is 
determined  by  article  357.  (2).  This  force  will  be  perpendicu- 
lar to  the  plane  passing  through  the  axis  and  the  centre  of  gravity, 
(3.)  Its  distance  from  the  axis  (^always  the  same,  whatever  be  the 
forces  and  their  directions^  will  be  equal  to  the  sum  of  the  products 
of  the  several  particles  of  the  body  into  the  squares  of  their  distances 
respectively  from  the  axis,  divided  by  the  product  of  the  mass  of  the 
body  into  the  distance  of  the  centre  of  gravity  from  this  same  axis. 

359.  V  denoting  always  the  velocity  with  which  a  determinate 
point  M  of  the  body  L  tends  to  turn  in  virtue  of  the  action  of 
any  number  of  forces,  or  of  their  resultant  q,  if  we  designate  the 
distance  of  any  particle   from  the  axis  of  rotation  by  r,  and   the 

mass   of  this  particle  by  m,  since  FM  :  v  :  :  r  :   -pr-p  we  shall 

have    -pYf  fo»'  the  velocity  of  rotation  of  the  particle  m,  and   -pirr 

for  the  force  it  would  exert,  and  consequently  for  the  resistance  it 
27.     would  oppose  to  g  by  its  inertia  ;  accordingly. 


Moment  of  Inertia.  235 

m  V  r  m  r'^  V 

-FmXt,  or  -^  60. 

will  be  the  moment  of  this  resistance ;  therefore  the  sum  of  the 
moments  of  these  resistances  which  the  particles  of  L  would  op- 
pose to  the  motion  of  rotation,    produced   by  g,   upon  these   par- 

tides,    is       „.,   ,  or   ^^^  f  m  r^,  for  the  two  expressions  are  the 

same,  since  v  and  FM  do  not  change,  whatever  be  the  particle  m, 
which  we  consider. 

We  hence  perceive,  that,  other  things  being  the  same,  the  re- 
sistance which  the  particles  of  a  body  oppose  to  the  motion  of 
rotation,  communicated  to  them,  is  so  much  the  greater  as  /  mr^ 
is  greater. 

The  quantity  -^^v  fmi^  is  called  the  moment  of  inertia  of 
a  body,  and  fmr^  the  exponent  of  the  moment  of  inertia. 

3G0.  We  shall  see  soon  how  the  exponent  of  the  moment  of 
inertia  in  any  body  may  be  determined ;  but  when  this  expo- 
nent has  been  determined  with  respect  to  any  axis  whatever,  it 
is  very  easy  thence  to  infer,  what  it  must  be  with  respect  to  any 
other  axis  parallel  to  the  former. 

Let  AB  be  any  axis,  and  A'B'  another  axis  parallel  to  it,  and  F'g- 170. 
passing  through  the  centre  of  gravity  G  of  the  body.  Let  m  be 
any  particle  of  this  body  ;  and  tlirough  m  suppose  a  plane  m  FF', 
perpendicular  to  the  two  axes  AB,  A'B'  -,  mF,  m  F',  being  drawn, 
and  the  perpendicular  m  P  being  let  fall  upon  FF^,  the  lines 
m  F,  m  F',  will  be  perpendicular  respectively  to  AB,  A'B'.  313!"" 

This  being  supposed,  we  shall  have, 

m^"=  vTf  -\-  FF'  -\-  2  FF  X  FP  ; 
hence,  ^^p- 

fm  '  nUr  —  fm  •  7n~F^  -\-Jm  •  FF  +Jm     2  FP  X  FP. 

Now,  since  the  distance  FF'  is  always  the  same,  whatever  be  the 

2  2 

particle  m  under  consideration,   j  m  •  FF'  is  simply  FF'  •  j*m,  or 


236  Dynamics. 


-2 


FF'  X  L,  the  mass  of  the  body  being  represented  by  L.  For 
the  same  reason/ wi  X  2  FF'  X  FP  is  simply  2  FFfm  X  FT. 
But  Tm  X  F-P,  being  the  sum  of  the  products  of  the  particles 
into  their  respective  distances  from  a  plane  passing  through  A'B', 
64.  that  is,  through  the  centre  of  gravity,  must  be  equal  to  zero  ;  we 
have  therefore  simply, 

2  2  2 

Jm  •  m  F  =fm  -  m  F  +  L  X  FF. 

2 

Hence,  knowing  the  exponent  C  m  '  m  Y'  of  the  moment  of  inertia 
with  respect  to  an  axis  passing  through  the  centre  of  gravity,  we  have 
the  exponent  with  respect  to  any  other  axis  parallel  to  this,  by  adding 
to  the  first  the  product  of  the  mass  into  the  square  of  the  distance 
between  the  two  axes. 

354.  From  this  result,  and  the  expression   for  the  velocity   of  rota- 

tion, it  may  be  inferred  that  of  all  the  axes  about  which  a  body  may 
be  made  to  turn  in  virtue  of  any  force  or  impulse,  those  about  which 
the  velocity  of  rotation  will  be  the  greatest  are  such  as  pass  through 
the  centre  of  gravity ;  since  the  exponent  of  the  moment  of  inertia 
with  respect  to  an  axis  passing  through  the  centre  of  gravity,  is 
less  than  it  is  with  respect  to  any  other  axis. 


Cf  the  Centre  of  Percussion  and  the  Centre  of  Oscillation. 

361.  The  foregoing  propositions  will  be  found  to  be  of  the 
greatest  importance  in  many  inquiries  to  be  resumed  hereafter ; 
we  shall  confine  ourselves  for  the  present  to  the  use  that  may  be 
made  of  them  in  finding  the  centre  of  percussion,  and  centre  of 
Fig.l7i.05a7Zo^io?2,  of  bodies  that  admit  only  of  a  rotation  about  a  deter- 
minate axis  or  point.  We  understand  by  the  centre  of  percus- 
sion, the  point  O  of  the  straight  line  FG,  where  it  would  be  ne- 
cessary to  place  a  body  in  order  that  it  might  receive  the  greatest 
impression  from  the  body  L  turning  about  jP.  Now  it  is  evident 
that  this  point  must  be  that  through  which  passes  the  resultant 
of  the  motions  of  rotation  of  all  the  particles  in  L.  The  centre  of 
percussion,  therefore,  is  determined  by  the  proposition  of  arti- 
cle 358. 


Centres  of  Percussion  and  Oscillation.  237 

As  to  the  centre  of  oscillation,  it  is  the  point  O  of  a  body  L  or  Fig.iTi. 
system  of  bodies,   whose   distance  from  F  is  equal  to  the   length 
which  a  simple  pendulum  must  have  in  order  to  perform   its  oscilla- 
tions in  the  same  time.     We  shall  see  that  this  centre  is  the   same 
as  the  centre  of  percussion. 

Indeed,  when  the  question  relates  to  gravity,  the  force  ^,  re- 
sulting from  the  action  of  gravity,  exerted  upon  each  material 
particle  of  a  body,  is  equal  to  the  whole  mass  multiplied  by  the 
velocity  communicated  by  gravity  in  an  instant  to  each  particle ; 
that  is, 

Q  =  u  X  L, 

u  representing  this  velocity.  Moreover,  this  resultant  q  passes 
through  the  centre  of  gravity  ;  and  consequently  its  perpendicular 
distance  from  the  fixed  point  F,  or  from  the  axis  passing  through  F, 
is  FH ;  hence  the  velocity  of  rotation  v,  of  any  point  Al,  when  the 
body  is  left  to  the  action  of  its  gravity,  is  354. 

J  m  r^  ' 

so  that,  for  the  centre  of  gravity  G,  the  velocity  is 

GG'  =  "X/'X/^  X  FG. 
J  m  r^ 

Now  in  order  that  a  simple  pendulum,  whose  length  is  FO, 
may  make  its  oscillations  in  the  same  time  with  the  body  L,  it 
is  necessary,  L  being  supposed  to  be  drawn  from  a  vertical  po- 
sition by  the  same  angular  quantity,  that  the  velocity  impressed 
by  gravity  at  O  (fig.  172),  perpendicularly  to  FO,  should  be  the 
same  as  that  of  the  point  O  (fig.  171);  in  other  words,  that  it 
should  be  to  the  velocity  of  G  (fig.  171),  as  FO  is  to  FG.  Now 
by  decomposing  the  velocity  u  or  OP  (fig.  172),  communicated 
by  gravity  in  an  instant  to  a  free  body,  into  two  others,  namely 
OK  in  the  direction  of  FO,  and  00'  perpendicular  to  FO,  we 
shall  have 

u  :  00'  :  :  FO  :  OZ  :  :  FG  :  FH; 

wlience 

u  :  00'  :.  FG  :  FH, 

and  consequently 


238  Dynamics. 

00'  =  ^-X  ™ 
FG 

We  hence  derive  the  proportion, 

FG  J  mr-^  ' 

which  gives 

±U  —  u  X  J^tl   X  ux  Lx  FHx  FG  ~  L  x  FG' 
357.    which  is  the  same  as  the  expression  for  the  centre  of  percussion. 

362.  Since  all  the  forces  which  act  upon  the  body  L,  or  upon 
a  system  of  bodies  that  admit  only  of  a  motion  of  rotation  about  a 
point  or  a  fixed  axis,  cause  in  this  body  such  a  velocity  that,  for 
any  given  point  M,  we  have 

v=^^XFM, 

J  m  ?"* 

and  since  it  is  evident,  that  if  this  body  were  to  turn  in  the  op- 
posite direction  with  the  same  velocity,  there  would  be  an  equi- 
librium among  all  these  forces ;  we  infer,  that  if  a  body,  turning 
with  a  velocity  which  for  a  determinate  point  M  is  equal  to  v, 
would  have  its  motion  counterbalanced  by  a  power  q,  the  direc- 
tion of  which  passes  at  a  distance  from  F  equal  to  D,  this  power 
taken  in  connexion  with  its  distance  J),  must  be  such  that  the 
moment  q  X  D  shall  be  equal  to  the  velocity  of  the  point  M,  di- 
vided by  the  distance  FJ\1,  and  multiplied  by  the  sum  of  the 
products  of  the  particles  into  the  squares  of  their  distances  re- 
spectively from  F,  or  from  the  axis  passing  through  F.  Indeed 
this  power  must  be  such  as  will  be  sufficient  to  produce  the 
same  velocity  in  the  body  L,  supposed  at  rest ;  and  this  velocity 
would  be 

V='-^XFM, 


which  gives 


?  X  D  =  vmS'^  '^ 


Centres  of  Percussion  and  Oscillation,  239 

3G3.  If  a  body  L,  of  any  figure  whatever,  admitting  only  of  a  Fig.  173. 
motion  about  a  fixed  point  F,  or  about  an   axis  passing  through  F, 
vvliich  may  in  other  respects  be  situated  as  we  choose,  if,  I  say,  a 
body  L  be  struck  by  a  body  N,  the  motion  of  each  after  collision 
may  be  determined  by  the  principles  above  established. 

Thus,  let  u  be  the  velocity  of  N,  before  collision,  according  to 
the  perpendicular  TH,  and  u'  its  velocity  after  collision  ;  u  —  u' 
will  be  the  velocity,  and  N  [u  —  u^)  the  quantity  of  motion,  lost 
by  collision,  and  which  will  pass  into  the  body  L.  This  quantity  of  290. 
motion  will  cause  in  Z*  a  velocity  of  rotation,  such  that  the  point  T, 
for  example,  will  turn  with  a  velocity 

N{u-u')xFH 

FII  being  drawn  perpendicular  to  TH.  Let  the  infinitely  small 
arc  TT',  described  about  the  centre  jP,  represent  this  velocity  ; 
the  parallelogram  TA  T'C  being  formed  upon  the  tangent  TA 
and  the  perpendicular  TH,  it  will  be  seen,  by  substituting  for 
TT  the  velocities  TA,  TC,  that  the  velocity  TA  cannot  afFect 
the  velocity  u'  which  the  body  N  must  have ;  but  that  the  velo- 
city TC  would  impair  the  velocity  u'  if  it  were  smaller  than  u' ; 
accordingly,  since  we  suppose  ihat  u'  is  actually  the  velocity 
which  N  preserves  after  collision,  it  is  necessary  that  TC 
should  be  equal  to  u'.  Now  the  similar  triangles  FHT,  TCV, 
give, 

FT  :  FH  :  TT    or    v  :  TC, 


354. 


whence, 


and  consequently 


u'  X  FT 
~^FH~' 


Substituting  for  v  this  value  in  the  equation  (i),  we  have 
W  X  FT  _  N{u  —  u')  X  FII         ,„, 

from  which  we  deduce  the  value  of  u* ;  thus, 


240  Dynamics, 

u'         N  X  u  X  FH       N  Xu  X  FH 

FH  ~  fmr^  fmr^         ' 


2 

u   X  /  m  r2  N  X  u  X  FH  __  N  X  u  X  FH 

fm  r^  X  FH  "*"    Jmf^  X  FH  ~         f  m  r^         ' 


lfmr^-\-NxFH\  _  N  Xu  X  FH 


( 


^'  V     /'«  r"^  X  FH      J  ~  fm  r2 

,  _N  XuX  FH  fmr^X  FH 

•^  fmr^^NxFH 


—      N  Xu  X  FH 
fmr^  +iVX  FH 


-2 


From  this  the  value  of  v,  or  the  velocity  of  rotation,  is  readily  ob- 
tained.    But  the  equal 
it  gives  the  proportion 


u  X  FT 

tained.     But  the  equation  v  =  — jprr     i  or  v  FH  =  u'  FT,  since 


u'  :  V  ::  FH  :  FT, 

makes  it  evident,  that  u'  is  the  velocity  of  rotation  of  the  point  H', 
from  which  it  will  be  seen,  that  the  point  H  turns  with  the  velocity 
that  remains  to  N  after  collision. 

364.  We  hence  perceive,  that  in  order  to  find  the  motion  of 
bodies  that  turn  about  a  fixed  point  or  axis,  we  must  be  able  to 
determine  the  value  o(  f  m  r^.  This  will  always  be  easy,  as  we 
shall  soon  show,  when  the  bodies  are  such  as  admit  of  being  ex- 
pressed by  equations.  We  may,  indeed,  in  any  case  consider  the 
body  as  composed  of  parallelopipeds,  pyramids,  &c.,  which  are 
capable  of  being  thus  expressed  ;  and  finding  for  each  component 
part  the  value  oi  J*  m  r^,  take  the  sum  of  these  as  the  total  value 
of  fm  r^  for  the  entire  body  or  system  of  bodies. 

When  the  body  is  such  as  admits  of  being  expressed  by  an 
equation,  we  proceed  thus  in  finding  the  value  oi  C  m  r^. 


Centres  of  Percussion  and  Oscillation.  24 1 

Let  AB  be  the  axis  of  rotation,  and  through  AB  suppose  two 
planes  -PQ,  AR,  to  pass  perpendicularly  to  each  other ;  lei  m  be  Fig.  174. 
any  particle  of  the  body  in  question,  and  having  let  fall  the  per- 
pendicular m  F  upon  AB,  we  draw  m  H  perpendicularly  to  the 
plane  RA;  and  joining  FH,  this  line  will  be  perpendicular  to 
AB,  and  consequently  to  the  plane  PQ.  The  right  angled  trian- 
gle m  HF  gives  Geom. 

IS6. 

Fin  =  FH+  H^; 
whence, 

f*m  X  F  m  or  Cm  r^  =  fm  X  FH  -{-  fm  •  H m. 

The  problem,  therefore,  reduces  itself  to  finding  the  sum  of  the 
products  of  the  particles  into  the  squares  of  their  distances  fronn 
two  planes,  which  pass  through  the  axis  of  rotation,  and  are  per- 
pendicular to  each  other.  Now,  when  the  algebraic  expression 
for  this  sum  is  found  with  respect  to  one  of  the  planes,  it  is  easily 
obtained  with  respect  to  the  other.  Let  us  therefore  inquire  how 
we  can  find  the  sum  of  the  products  of  the  particles  of  a  body 
into  the  squares  of  their  distances  respectively  from  a  known 
plane. 

We  will  suppose  the  body  divided  into  infinitely  thin  strata, 
parallel  to  the  given  plane ;  and,  representing  the  thickness  of 
one  of  these  strata  by  DD',  its  surface  by  a,  and  its  distance  FD  F'g- 175. 
from  the  plane  in  question  by  x,  since  the  points  of  the  surface  a 
are  all  distant  from  the  plane  PQ  by  the  same  quantity  x,  we 
shall  have  oc^  a  d  x  for  the  sum  of  the  products  of  all  the  points 
of  this  surface  into  the  squares  of  their  distances  respectively 
from  this  plane,  and  consequently  fx^  o  d  x  for  the  entire  sum 
of  these  products  for  the  whole  body. 

If  we  represent,  in  like  manner,  by  a/  the  corresponding  dis- 
tances from  the  plane  perpendicular  to  PQ,  and  passing  through 
the  axis  of  rotation  AB  (the  body  being  supposed  to  be  divided 
into  strata  parallel  to  this  second  plane),  and  by  a'  the  surface 
of  one  of  these  strata,  we  shall  have  f  x''^  a'  d  x'  for  the  sum  of 
the  products  of  the  particles  into  the  squares  of  their  distances 
respectively  for  this  second  plane  ;  and  accordingly 
Mech.  31 


242  Dynamics. 

far^adx  +/a/2  „'  d  x' 

will  be  the  value  of  the  sum  of  the  products  of   each  particle  of 
the  body  into  the  square  of  its  distance  from  the  axis  AB. 

365.    Let  us  now   suppose,   by  way  of   illustration,   that  the 
"'      '  body  in   question    is    a   rectangular   parallelopiped,    turning   about 
the  axis  AB  perpendicular  to  the  axis  of   the   parallelopiped,   and 
to  the  side  IK.     By  the  nature  of  this  body,  the  surface   u  is  con- 
stant ;    thus  the  integral  f  x^  a  d  x  \s  -77-,    which,   when     x    is 

equal  to  the  altitude  h  of  the  parallelopiped,  becomes  -5-. 

In   like   manner,  o'  being  a  constant  quantity,  fx"^  a'  d  x/  be- 
comes 

"3"' 

or,  MN  being  represented  by  A',  which  gives  x'  =.  ^  h', 

8  ^       3     ' 

and,  as  the  plane  which  passes  through  the   axis  divides  the   body 
into  two  equal  parts,  the  two  halves  will  be 

*  ^      3      °'"   "l2~ ' 

therefore  the  entire  sum  of  the  products  will  be 

h^  a  _.    IP  a' 
"3~  "^  "l2~' 

357.  If  we  would  find  the  centre  of  percussion  or  of  oscillation,  we 

have  only  to  divide  this  quantity  by  the  product  of  the  mass  of  the 

parallelopiped  into  the   distance   of  its  centre  of  gravity  ;  that   is, 

Geom.    hy  h  h'  f  X  \  h  or  \  h^  h'  /,  LM  being  denoted  by  /,  which  gives 

for  the  distance  of  the  centre  of  percussion  or  of  oscillation 

3  A2  h'f  '^  6h^h'f  °^    T  +  6~A' 
siDce 

a  =  h\f,     and     a'  =.  hf. 


Centres  of  Percussion  and  Oscillation.  243 

If  h'  is  very  small  with  respect  to  h,  so  that  tt-,  may  be  neg- 

2  h 

lected,  the  expression  becomes  -^.     Hence,    the   distance    of   the 

centre  of  percussion  or  centre  of  oscillation  of  a  straight  line,  or  of 
a  parallelogram,  turning  about  one  of  its  sides,  as  an  axis,  is  |  of 
the  length  from  the  point  of  suspension  or  axis. 

Thus,  the  rod  or  bar  F^,  turning  about  the  fixed  point  F  would  Fig.  177. 
strike  a  nail  T  with  the  greatest  effect  when  the  distance  of  the  nail 
FP  is  equal  to  |  FA. 

If  the  rod  FA  be  considered  as  turning  by  the  action  of 
gravity  only,  the  force  which  it  would  exert  upon  the  nail, 
would  be  equal  to  the  mass  of  the  rod  multiplied  by  the  velocity 
acquired  by  the  centre  of  gravity  G,  in  falling  along  G'G,  that 
is,  by  the  velocity  acquired  by  a  heavy  body  in  falling  through  the 
height  GD.  339. 

366.  We  take  the  sphere  as  a  second  example.     In  this  caseFig.l78. 
the  surface  which  we  have  called  a,  is  a  circle,  having  for  its  radius 
LM,  which  I  shall  call  y ;  and,  n  being  the  circumference  of  a  cir- 
cle whose  diameter  is  1 ,  we  have 

„  ,,2  —  ,^  Geoin. 

ny    -a.  291. 

Let  DI  be  denoted  by  z,  and  the  radius  of  the  sphere  by  r  ;  we 
have 

y^  =:2rz  —  z^,  Trig. 

^  '  101. 

and  consequently, 

(/  =  n{2RZ  —  z^). 

Calling  DF,  a, 

FI 01  X  z=  z  -\-  a,     and     d  x  =i  d  z  ; 

consequently, 

Jar^  a  d  x 

becomes 

/(~  +  a)2  Xn{2Rz  —  z'-)dz, 


244  ■  Dynamics. 

or,  by  developing  the  whole, 

fii{2a^Rzdz-\-iaRz^dz  —  a^z^dz  +  2Rz^dz  —  ^az^dz—z^dz), 

Cai.  85.  and  by  integrating,  we  have 

which,  when  z  =  2  r,  becomes 
TT  (4  a^  r3  +  \2  a  r4  —  f  a^  r3  +  8  r^  —  8  a  r^  —  V  R^) 

or 

7t  (A  a2  r3  -I-  ^  a  r4  4-  I  r5). 

To  find  the  value  of  C  a/^  a  d  x\  it  is  not  necessary  to  begin  the 
calculation  again,  since  from  the  regular  figure  of  the  sphere,  it 
is  evident  that  this  value  will  be  similar  to  the  form<;r ;  we  have 
only  to  suppose,  therefore,  that  a,  which  expresses  the  distance 
of  the  plane  PQ  from  the  surface,  becomes  —  r;  that  is,  that 
this  plane  passes  through  the  centre,  it  being  supposed  at  the 
same  time  to  be  perpendicular  to  its  first  position,  and  we  shall 
have 

TT  (A  r5  —    I  r5  _|_  I  r5)  or  71    X     tV  R^- 

The  two  integrals  being  added  together,  make 

«  (f  a^  r3  +  I  a  r4  +  \%  R=). 

Qgon,^     Since  the  bulk  of  the  sphere  is  tt   X    f  R^  or  |  u  r^,  and  the   dis- 

548-       tance  of  its  centre  of  gravity  from  the  plane   PQ  is  a  -(-  r,  if  we 

divide  the  above  result  by  the   product  f  ti  r^  X  (a  +  r)  of  these 

two  quantities,  we  shall  have  the  distance  of  the  centre  of  oscillation 

S57,    and  that  of  percussion  •,  thus, 

YQ  ^  TT  (4  g^  r3  +  I  fl  R^  +  H  ^') 
I  ;r  r3  (a  4-  r) 


Centres  of  Percussion  and  Oscillation.  245 

_     g^  -f  2  a  R  +  1  R^ 


a2  +  2  a  u  +  r2  + 


Hence  the  centre  of  oscillation  and  that  of  percussion  are  below  the 
centre  of  the  sphere ;  and  the  centre  of  the  sphere  cannot  be 
taken  for  the  centre  of  oscillation  or  that  of  percussion,  except 
when  its  radius  is  very  small  compared  with  the  distance  of  the 
centre  G  from  the  point  of  suspension. 

If  the  sphere  is  suspended   by  a  rod  or  lamina,  and  we  would 
have    regard   to   its    mass,   it   will   be    recollected,   that   we    have    365. 

h^  (J  h'^  (J 

found  —^ —  -|~     io     ^'^'^  ^^^  2'J'^  o^  ^^   products  of  the   particles 

of  such  a  body  into  the  squares  of  their  distances  respectively  from 
the  fixed  point  or  axis.  Now  h  is  what  we  have  represented  by  a ; 
moreover,  since 

a  =L  h'  f,  and  a'  =  h  f=z  a  f, 

we  shall  have  by  substitution, 

a^h'f         h'^af , 
3      "*"       12     ' 

this  quantity  and  that  for  the  sphere  must  be  multiplied  respec- 
tively by  the  specific  gravities  S,  <S',  of  these  bodies,  if  their  spe- 
cific gravities  be  different ;  then  by  adding  the  two  products,  we 
shall  have, 

^-^^+  ^^tI^+  ^'  "  (^ "'  ^'  +  ^  ^  ^'  +  '^  ^') 

for  the  sum  of  the  products  of  the  particles  of  the  whole  system 
into  the  squares  of  their  distances  respectively  from  the  axis.  This 
sum  divided  by  the  product  of  the  masses,  S  a  h'  f  -\-  S'  |-  tt  b^ 
into  the  distance  of  the  centre  of  gravity  from  the  axis,  gives  the 
distance  of  the  centre  of  oscillation. 


246  Dynamics. 

367.  It  may  suffice  in  practice  to  divide  the  body  into  a  great 
number  of  parts,  and  multiply  each  part  by  the  square  of  its  distance 
from  the  axis  in  order  to  obtain  with  sufficient  exactness  the  value 
of  J^m  r^. 


Of  the  actual  Length  of  the  Seconds  Pendulum. 

368.  The  number  of  vibrations  performed  in  the  same  time 
by  two  different  pendulums,  urged  by  the  same  gravity,  being 
inversely  as  the  square  roots  of  the  lengths  of  these  pendulums, 
we  can  find  very  nearly  the  length  of  the  seconds  pendulum  for 
any  given  place  by  a  very  simple  process.  Having  suspended 
to  a  very  fine  wire  of  at  least  three  feet  in  length,  a  small  dense 
body,  as  a  ball  of  lead;  gold,  or  platina,  we  ascertain  the  length 
of  this  wire  and  the  radius  of  the  ball  with  great  exactness.  We 
then  cause  this  pendulum  to  vibrate  by  drawing  it  a  litde  from  a 
vertical  position,  and  count  the  number  of  vibrations  performed  in 
a  given  time,  as  one  hour,  very  carefully  determined,  and  then 
make  use  of  the  proportion  ;  as  3G00,  the  number  of  vibrations  to 
be  performed  by  the  pendulum  sought,  is  to  the  number  actually 
performed  by  the   above   pendulum,   so  is  the  square  root  of  the 

346,  length  of  this  latter  pendulum  to  a  fourth  term  or  x,  which  will  be 
the  square  root  of  the  length  of  the  pendulum  sought ;  and  by 
squaring  this  fourth  term,  we  shall  have  very  nearly  the  length  of 
the  pendulum  required  to  vibrate  seconds.    /^^  . 

This  result  would  be  exact  only  on  the  supposition  that  the 
wire  or  string  is  without  weight,  and  that  the  ball  consists  only 
of  a  single  particle  or  has  its  matter  concentrated  at  the  centre. 

369.  If  we  attempt  to  find  geometrically  the  centre  of  oscil- 
lation of  the  ball  and  wire,  we  shall  still  be  liable  to  some  small 
error  arising  from  irregularities  in  the  form  and  distribution  of 
the  matter  in  question.  We  accordingly  have  recourse  to 
another  method,  depending  on  a  curious  property  of  the  com- 
pound pendulum  by  which  the  distance  between  the  point  of  sus- 
pension and  centre  of  oscillation,  answering  to  the  length  of  the 
simple  pendulum  vibrating  in  the  same  time,  can  be  ascertained 
with  the  greatest  precision. 


Length  of  the  Seconds  Pendulum.  247 

We   have  obtained   a   general  expression  for  .the  distance  in 
question,  as  follows,  namely,  Fio-.i7o. 


— Q 
jPf^ fmr^ C  m  '  m.F 

~  L  X  FG  ~  L  X  FP' 


But 


Jm  •mF=rm-mF'-\-Lx  FF' ; 
whence,  by  substitution, 


that  is, 


p^  _fm-mF-{-LxFF^ 
~  L  X  FF'  ' 


FO  or  FF  +  F0  =  ^^^^  +  FF, 


whence 

FO 


2 

f  m  •  m  F 

~  L  X  FF' 


Thus,  the  distance  of  the  centre  of  oscillation  below  the  centre  of 
gravity  is  equal  to  the  sum  of  all  the  parts  multiplied  by  the  squares 
of  their  respective  distances  from  the  axis  drawn  through  the  centre  of 
gravity,  divided  by  the  product  of  the  mass  into  the  distance  of  the 
centre  of  gravity  from  the  aocis  of  suspension. 

Now  by  multiplying  both  members  of  the  above  equation  by 
FF',  and  dividing  both  by  jPO,  we  shall  obtain, 

2 

„,„_  fm-mF 
L  X  OF' 

Accordingly,  if  we  consider  the  body  as  inverted,  and  make  O  the 
point  of  suspension,  F  will  become  tlie  centre  of  oscillation,  since 
we  have  the  same  expression  as  before  for  the  distance  of  this  point 
below  the  centre  of  gravity. 

We  hence  infer,  that  the  point  of  suspension  and  centre  of  oscil- 
lation are  convertible,  that  is,  either  being  made  the  point  of  suspen- 
sion the  other  becomes  the  centre  of  oscillation. 


361. 


360. 


248  Dynamics. 

Reciprocally,  if  two  points  are  so  chosen,  or  so  adjusted  to 
each  other  by  movable  weights,  that  the  pendulous  body  shall 
vibrate  in  the  same  time  when  suspended  from  one  as  when  sus- 
pended from  the  other,  these  points  are  alternately  the  centres  of 
oscillation  and  points  of  suspension,  and  the  distance  asunder  is  the 
length  of  the  pendulum  in  question,  and  equal  to  that  of  a  simple 
pendulum  vibrating  in  the  same  time.  The  above  proposition  was 
demonstrated  by  Huygens,  the  original  author  of  the  theory  of  the 
pendulum,  but  it  was  not  till  very  lately  applied  to  any  useful  pur- 
pose. Captain  Kater  was  the  first  to  perceive  that  it  furnished  a 
very  simple  and  accurate  method  of  determining  the  length  of  the 
compound  pendulum. 

Figure  179  represents  Captain  Kater's  pendulum.  The  axes 
JP,  O,  were  adjusted  by  means  of  intermediate  movable  weights 
C,  D,  and  with  so  much  accuracy  that  the  number  of  oscilla- 
tions made  in  twenty-four  hours,  F  being  uppermost,  differed  from 
those  performed  in  the  same  time  with  O  uppermost,  less  than 
half  a  vibration  ;  and  the  means  of  twelve  sets  of  observations  with 
first  one  then  the  other  uppermost,  differed  from  each  other  less 
than  the  hundredth  of  a  vibration.  The  length  of  the  pendulum, 
as  thus  obtained,  is  stated  to  be  39,1386  inches.  This  is  for  the 
latitude  of  London,  or  51°  31'  08'',04  iV.,  and  on  the  supposition 
of  the  arcs  of  vibration  being  infinitely  small,  taking  place  in  a 
vacuum,  and  at  the  level  of  the  sea,  the  temperature  being  62°  by 
Fahrenheit's  thermometer.  Tliis  determination  exceeds  what  was 
considered  the  most  accurate  result  of  the  methods  previously  in 
use  by  0,00813  or  nearly  one  hundredth  of  an  inch,  a  very  im- 
portant difference  in  researches  where  the  ten-thousandth  of  an  inch 
is  an  appreciable  quantity. 

It  may  be  observed,  moreover,  that  if  the  two  axes  of  the 
pendulum  be  cylindric  surfaces,  the  points  of  suspension  and 
oscillation  are  truly  in  these  surfaces,  and  the  length  sought  is 
rigorously  the  distance  between  these  surfaces.  This  second 
property,  so  necessary  to  the  completeness  of  the  method,  when 
actually  applied  to  practice,  was  discovered  by  Laplace.  See 
Ed.  Rev.  vol.  xxx.  p.  407.     Phil.  Trans,  for  1818. 


Force  of  Gravity.  249 

370.  It  is  not  necessary  to  go  through  the  same  process  in  order 
to  find  ti-ie  length  of  a  penduhim  required  to  vibrate  in  any  other 
proposed  time,  as  half  a  second,  or  half  a  minute.  The  principles 
we  have  investigated  will  enable  ns  to  solve  all  problems  of  this 
kind  with  the  greatest  facility  and  exactness,  when  the  length  and 
time  of  vibration  of  one  pendulum  is  known.  Thus  if  it  is  pro- 
posed to  find  the  length  of  a  pendulum  required  to  vibrate  half  ^^^• 
minutes,  the  proportion 

t^  u  t'-  ::  a  :  a', 

by  substituting  for  t,  t',    I"  and  30'',  and  for  a  39,1386,  the  length 
of  the  seconds  pendulum,  we  have 

12  :  :  (30)2   .  .   39^1386  :  a'  —  39,1386   X   900  =  35224,74 
inches,  or  2935,39  feet. 

In  like  manner,  the  length  and  time  of  vibration  of  one  pen- 
dulum being  known,  the  time  of  vibration,  in  the  same  place,  of 
any  other  pendulum  whose  length  is  given,  may  be  determined. 
Suppose,  for  example,  that  it  is  required  to  find  the  time  in  which 
a  pendulum  of  20  feet,  or  240  inches  in  length,  would  perform  its 
vibrations  ',  by  substituting  the  known  quantities  in  the  general  pro- 
portion, 

\^a  '  \/a'  :  :  t  :  t', 

we  have 

-\/3iM386  :  V250  :  :  I"  :  t'  =     I    ^^^      =  2'^5  nearly. 


Measure  of  the  Force  of  Gravity. 


371.  It  will  be  easy  now  to  determine  through  what  space  a 
heavy  body  must  pass  in  the  first  second  of  its  fall,  the  air  and  all 
other  obstacles  being  removed.     For  the  equation  342. 

i'  = 


gives,  by  squaring  both  members  and  transposing, 

71"  a 

Mech.  32 


250  Dynamics. 

in  which  g  represents  the  velocity  acquired  by  a  heavy  body  at 
the  end  of  the  first  second  of  its  fall,  and  which  is  double  the 
height  or  space  through  which  it  would  descend  in  a  second  from 
266.  a  state  of  rest ;  a  is  the  length  of  the  pendulum  each  of  whose 
vibrations  is  performed  in  the  time  f.  Accordingly,  if  for  t'  we 
put  one  second,  a  must  be  39,1386  inches  for  the  latitude  of  Lon- 
don.*    jNIoreover  n,  the   ratio  of  the  circumference  of  a  circle  to 

Geom.    its  diameter,  is  equal  to  3,1416  nearly  :  hence 
294.  ^ 


Accordingly, 


g  =  (3,1416)2  X  39,1386. 


3,1416....2  log....0,99430 
39,1386 log.. ..1,59260 


386,28  2,58690 


The  value  of  g,  therefore,  is  386,28  inches,  or  32,1 9f  feet,  equal 
to  32,2  nearly;  and  half  this  quantity  or  16,1  is  the  space  de- 
scribed by  a  heavy  body  in  an  unresisting  medium  at  the  surface 
of  the  earth  in  one  second  from  the  commencement  of  its  motion. 
273.    We  have  thus  fulfilled  our  promise. 


Application  of  the  Pendulum  to   Time-Keepers. 

372.  The   pendulum   attached  to  clocks  for  the  purpose  of  reg- 
Fig.180.  ulating   their   motions,  is  ordinarily  a  rod  of  metal  or  wood   loaded 

*  The  length  of  the  seconds  pendulum,  and  consequently  the 
value  of  g,  is  referred  to  the  latitude  of  London  on  account  of  the 
great  accuracy  of  the  observations  that  have  been  made  at  this 
place.  The  difference,  however,  in  the  length  of  the  pendulum  in 
different  latitudes,  at  the  level  of  the  sea,  is  so  small  as  to  amount 
only  to  about  -i  of  an  inch  at  the  extreme,  or  when  the  places  to 
which  the  observations  relate  are  the  equator  and  the  pole  ;  and 
the  difference  in  the  value  of  g  at  these  places,  is  only  about  two 
inches,  as  may  be  easily  shown  by  the  above  formula. 

t  The  most  accurate  observations  on  the  length  of  the  seconds 
pendulum  at  Paris  in  latitude  48°  51'  give  for  the  value  of  g 
32,182  ft. 


Application  of  the  Pendulum  to  Time-Keepers.  251 

at  the  lower  extremity  with  a  weight  in  the  form  of  a  lens,  so 
placed  as  to  meet  with  as  little  resistance  as  possible  from  the  air. 
The  axis  also  or  point  of  suspension  is  fitted  to  have  very  little 
friction.  Connected  with  the  pendulum,  is  a  train  of  wheels  and 
pinions,  the  teeth  and  leaves  of  which  are  so  adapted  to  each 
other,  that  the  motions  correspond  to  the  several  divisions  of 
time,  and  their  axes  carry  indexes  that  show  by  the  arcs  they 
describe,  the  hours,  minutes,  and  seconds.  Around  tlie  axis  at 
one  extremity  of  this  train  of  wheels,  is  wound  a  cord  bearing  a 
weight,  that  would  put  the  whole  system  in  rapid  motion,  but  for 
the  appendage  to  the  pendulum  CFD  at  tbe  other  extremity  of  the 
train  of  wheels,  which,  while  the  pendulum  is  at  rest,  effectually 
prevents  all  motion.  But  if  the  pendulum  be  made  to  vibrate, 
it  will  suffer  one  tooth  to  pass  or  escape  at  each  vibration,  while 
at  the  same  time  the  impulse  of  the  teeth  upon  the  arms  FC,  FD, 
is  so  adjusted,  by  increasing  or  diminishing  the  weight,  as  just  to 
overcome  the  friction  and  the  resistance  of  the  air,  and  thus  to 
keep  up  the  motion,  while  the  action  of  the  weight  continues. 
The  contrivance  by  which  the  train  of  wheels  is  connected  with 
the  pendulum,  is  called  the  escapement. 

373.  On  account  of  the  constancy  of  gravity  the  oscillations 
of  the  pendulum,  other  things  being  the  same,  must  be  equal  or 
of  the  same  duration.  There  are,  however,  several  causes  that 
tend  to  disturb  this  isochronism.^(l.)  The  air  is  subject  to  changes 
of  density,  on  account  of  which  the  arcs  of  vibration  will  some- 
times be  longer  and  sometimes  shorter,  while  the  maintaining 
power  remains  the  same.*  But  if  these  changes  are  noted,  or  if 
the  arcs  of  vibration  are  noted,  the  deviation  from  perfect  regu- 
larity can    be  calculated,   and  allowance  made   accordingly. f     It 

*  As  the  air  becomes  more  dense  the  pendidum  is  more  resisted 
and  would  seem  to  be  retarded,  but  the  arc  of  vibration  being  di- 
minished, the  clock  goes  faster,  so  that  one  of  these  causes  tends  343^ 
to  counteract  the  other.  In  like  manner,  when  the  motion  of  the 
axis  and  wheels  is  obstructed  by  dust  or  want  of  oil,  the  impulse 
of  the  weight  communicated  to  the  pendulum  is  diminished,  the 
arcs  of  vibration  are  reduced,  and  hence  there  is  a  tendency  to 
increase  its  rate  of  going. 

t  If  the    pendulum   could    be    made  to  move   in   the   arc  of  a 
cycloid,  it  will  be  perceived  from  what  has  been  said  of  this  curve,     3.').3. 


252  Dynamics. 

may  be  remarked,  moreover,  that  the  irregularity  from  this 
cause,  is  rendered  for  common  purposes  altogether  inconsidera- 
ble, by  making  the  pendulum  very  heavy,  and  the  arcs  of  vi- 
bration very  small. 

374.  (2.)  A  much  more  important  source  of  error,  in  the  rate 
of  going  of  common  clocks,  is  to  be  referred  to  changes  in  the 
actual  length  of  the  pendulum  arising  from  heat  and  cold.  A 
brass  pendulum  rod,  for  iiistance,  has  its  length  increased  about 
two  hundredths  of  an  inch  for  a  change  of  temperature  of  30°  of 
Fahrenheit's  thermometer.  This  would  seem  to  be  a  small 
quantity  ;  yet  as  it  is  continually  exerting  an  influence,  the  accu- 
mulated effect  in  the  course  of  24  hours  or  8G400''  amounts  to 
more  than  a  third  of  a  minute.  The  expansion  of  iron  is  about 
I  of  that  of  brass.  There  are  some  kinds  of  wood  that  are  sub- 
ject to  very  little  variation  of  length,  particularly  in  the  direction 
of  the  fibres,  on  account  of  temperature.  Still  no  substance  is 
entirely  free  from  these  changes.  The  effect  of  any  augmenta- 
tion   or   diminution    of  length  in  the   pendulum   may   be  computed 

346.    by  means  of  the  principles  that  have  been  investigated. 

375.  But  we  can  obtain  more  convenient  and  sufficiently  ex- 
act formulas  for  the  variation  in  the  rate  of  the  going  of  a  clock, 
when  the  changes  in  the  length  of  the  pendulum  are  very  small, 
as  those  are  which  arise  from  heat  and  cold,  a  being  the  exact 
length  of  the  seconds  pendulum,  or  that  by  which  the  clock 
w'ould  keep  correct  time,  and  a'  the  actual  length,  as  affected  by 
heat  and  cold,  if  we  put  ?i  for  the  number  of  oscillations  in  a  day, 
performed  by  the  former,  n'  for  the  corresponding  number  of  the 
latter,  and  i'  for  the  time  of  this  latter,  we  shall  have 


346. 


348. 


also 


'\  a 


n' 


that  all  arcs  whether  longer  or  shorter,  would  be  described  in  the 
same  time.  But  the  practical  difficulties  attending  all  the  methods 
hitherto  proposed,  are  such  as  to  occasion  errors,  that  more  than 
compensate  for  the  theoretical  advantages  to  be  derived  from  a 
cycloidal  motion. 


Application  of  the  Pendulum  to  Time-Keepers.  253 

whence 


71'         "X  a 


and  .49^ 


\.  -JL . 


a'  = 


i'2  • 


Suppose  a;  to  be  the  augmentation  or  diminution  of  length  in  ques- 
tion, and  y  the  corresponding  daily  loss  or  gain  in  seconds,  we 
shall  have 


a'  =1  a  ±  X  =z  — TT-  = 


n 


/2 


(«  qp  y)2  -  n^  ^  2  «  3/  +  3/2       1  IF  2J> 


nearly,  neglecting  ^  as  very  small ;  that  is, 

/  2w\ 

a  zt:  X  ■=.  a  ii  ±-^), 

nearly,  from  which  we  obtain, 

2  ?/  a         J  n  X 

X  =  -^—,   and  y  =  ^r— . 

»  ^        2  a 

r 

Thus,  if  for  any  given  rise  of  the  thermometer,  the  pendulum 
is  lengthened   one  hundredth  of    an   inch,   we   shall  have  for  the 

number  of  seconds  lost  per  day,  , 

'  •        V 
nx        86400"  X  0,01         ,,„         , 
y  =  2-a  =  -2ir39ji~  =  ^^     "'''^^* 

On  the  other  hand,  if  a  clock  is  known  to  keep  time  correctly 
at  a  particular  temperature,  as  55°  for  instance,  and  at  32°  is  found 
to  gain  7"  a  day,  we  should  be  able  to  determine  the  corresponding 
diminution  in  length,  or  the  contraction  in  the  rod  of  the  pendulum, 
answering  to  this  number  of  degrees  ;  thus, 

2  ;/  a  __  2  X  7  X  39,14        .....     , 

X  =  =: Bi^,nr>^ •  :=  0,000  inches. 

n  80400 

370.  It  will  be  seen,  therefore,  that  by  rendering  the  weight 
of  the  pendulum  movable  upon  the  rod,  and  connecting  it  with  a 
micrometer  screw,  a  correction   may  be  appli  d  for  the  expansion 


/ 


254  Dynamics. 

and  contraction  according  to  the  state  of  the  thermometer.  But 
a  more  convenient  method  lias  been  devised  by  whicli  the  expan- 
sion of  one  metal  is  made  to  counteract  tliat  of  another.  The 
expansion  of  iron  and  brass  being  to  each  other  as  three  to  five, 

^'o-^^^-if  we  make  the  rod  Fi?  of  iron,  and  the  rod  AO  of  brass  in  the 
proportion  of  5  to  3 ;  they  being  connected  at  the  lower  extrem- 
ities, and  the  weight  being  attached  at  O,  the  rod  AO  will  expand 
upward  just  as  much  as  the  rod  FA  expands  downward,  and  the 
point  O  where  the  weight  is  applied,  will  consequently  remain 
amid  all  changes  of  temperature  at  the  same  distance  from  F, 
the  point  of  suspension.  A  number  of  rods  of  each  kind  is  usu- 
ally employed  as  represented  in  figure  182,  where  the  rod  which 
supports  the  weight,  is  attached  at  F  and  free  at  D,  D',  the  brass 
rods  expanding  upward  and  the  iron  ones  downward  as  before ; 
so  that  if  the  proper  proportion  as  to  length  be  observed,  a  com- 
pensation for  the  effect  of  temperature  will  be  obtained.  Other 
means  have  been  invented  for  accomplishing  the  same  purpose. 
Of  these  we  shall  mention  only  one  ;which  has  been  attended 
with  great  success.  The  weight  AB  is  made  to  consist  of  a  glass 
tube  about  two  inches  in  diameter,  and  from  4  to  7  inches  long, 

Fig.183.  filled  with  mercury.*  As  the  rod  of  the  pendulum  supporting  this 
weight,  expands  downward,  the  mercury  expands  upward,  as  in 
the  contrivance  first  mentioned,  and  the  quantity  may  be  increas- 
ed or  diminished  till  a  compensation  is  effected.  A  clock  pro- 
vided with  a  pendulum  of  this  construction,  made  by  T.  Hardy  of 
London,  for  the  Royal  Observatory  at  Greenwich,    was  found  after 

*  The  expansions  of  glass  and  mercury  being  as  1  to  10  very  near- 
ly, if  the  suspending  rod  be  of  glass,  the  column  of  mercury  must  be 
■^  of  the  length  of  the  pendulum  or  about  4  inches.  If  the  rod  be  of 
iron,  as  this  substance  has  a  greater  expansion  in  the  ratio  of  3  to  2 
nearly,  the  column  of  mercury  should  be  about  G  inches.  A  steel 
rod  would  require  a  column  6,4  inches  in  length,  which,  on  the  sup- 
position of  a  diameter  of  two  inches,  would  weigh  lOlbs.  From 
accurate  calculation,  it  is  found  that  if  such  a  pendulum  should  keep 
perfectly  true  time,  when  the  thermometer  is  at  30°,  and  that  it 
should  gain  or  lose  1"  a  day  when  the  thermometer  is  at  90°,  the  im- 
perfection would  be  remedied  by  the  subtraction  or  addition,  as  the 
case  required,  of  10  ounces  of  mercury. 


Application  of  the  Pendulum  to  Time-Keepers.  255 

two  years'  trial  to  vary  only  j  of  a  second  in  24  hours  from  its 
mean  rate  of  going.  A  clock  of  the  same  construction  owned  by 
W.  C.  Bond  of  Boston,  though  much  less  costly,  has  been  found 
by  careful  observation,  to  go  with  nearly  the  same  accuracy. 

377.  A  watch  or  chronometer  differs  from  a  clock  by  having 
a  spring  for  its  maintaining  power,  and  a  horizontal  instead  of  a 
vertical  pendulum,  in  which  a  small,  fine  spring  performs  the 
office  of  gravity.  The  pendulum  or  balance,  in  this  case  also,  Fig.184. 
is  subject  to  irregularity  from  heat  and  cold,  and  requires  a 
distinct  compensation.      Considerable  weights  m,  m',  are  attached 

to  the  balance  by  means  of  slips  C  m,  C  m',  of  brass  and  steel, 
the  brass  slip  in  each  being  outermost.  While,  therefore,  the 
general  expansion  of  the  wheel  tends  to  throw  the  weight  to  a 
greater  distance,  the  superior  expansion  of  the  brass  slip  over  the 
steel  brings  the  weight  nearer  to  the  centre,  and  the  length  of  the 
slip  being  properly  adjusted  to  the  weight,  the  centre  of  oscillation, 
or  rather  of  gyration,  will  be  preserved  always  at  the  same  distance 
from  the  axis. 

378.  We  have  found  formulas  for  the  difTerence  in  the  rate 
of  going  of  a  clock  answering  to  small  changes  in  the  length  of 
the  pendulum,  the  position  with  respect  to  the  centre  of  the  earth, 
and  consequently  the  force  of  gravity,  being  supposed  to  remain 
the  same.  It  will  be  easy  also  to  find  formulas  for  the  varia- 
tion in  the  force  of  gravity  and  in  the  rate  of  the  going  of  a 
clock,  depending  upon  small  changes  of  distance  from  the  centre 
of  the  earth,  n,  n',  for  example,  being  the  number  of  vibrations 
of  the  same  pendulum  at  the  two  stations  respectively,  the  pen- 
dulum being  supposed  to  vibrate  seconds  at  the  first  j  from  the 
proportion, 


la'         I  rt 


when  a'  =  a,  we  have 


g'  =  '- 


17  n 


'2 


If  the  second  station  be  below  the  first,  or  that  at  which  the 
pendulum  vibrates  seconds,  g*  will  exceed  g,  and  the  clock  will 


348. 


256  Dynamics. 

gain  ;  on  the  contrary  supposition  it  will  lose.     Let 

and  let  y  denote  the  daily  gain  or  loss  in  seconds  ;  we  shall  have 
„(^  -t-  ^N  _    g  (^^  ^  yy  _  g  (»^  ±  2  y  n) 

nearly.     Whence, 

X  =■  ±  — ^. 

Thus,  if  a  pendulum  fitted  to  vibrate  seconds  at  the  equator, 
would,  upon  being  carried  to  the  pole,  gain  5'  or  300''  a  day,  we 
should  have 

_  2  X  300  _  J_^ 
^  ~    86400    ~  144 ' 

that  is,  the  force  of  gravity  at  the  equator  is  to  that  at  the  pole,  on 
this  supposition,  as  144  to  145. 

Let  the  difference  h  in  the  distances  of  the  two  stations  from 
the  centre  of  the  earth  be  given,  gravity  being  supposed  to  vary 
inversely  as  the  square  of  the  distance,  the  gain  or  loss  of  the  clock 
might  be  readily  found  as  follows. 

If  we  call  R  the  distance  of  the  centre  of  the  earth  from  the 
first  station,  and  g  the  force  of  gravity  at  this  station,  the  pendu- 
lum being  supposed  to  vibrate  seconds,  we  shall  have  for  the  dis- 
tance of  the  second  station  R  ±  h,  and  for  the  force  of  gravity  at 
this  station. 


R2  ,,         2h\ 


2  h  . 

nearly.    Hence,  putting  -^  for  x  in  the  above  formula,  we  obtain 

2  h  ,    2  y        ,    ,  nh 

=^-R='^-^'^'''^^y  =  =^-R' 

Thus,  if  the  second  station  be   above  the  first,  as   1    mile  for  in- 
stance, the  radius  of  the  earth  being  3956,  or  4000  nearly,  the 


Rotation  of  Bodies  unconfined.  257 

the  formula  gives 

n,    86400    91    fi// 

—  y —  40  0  0  — -^ijO  , 

the  sign  —  indicatins  that  the  cloclc  loses. 

Rotation  of  Bodies  unconfined. 

379.  It    has  been   demonstrated    that  when   a  body   L  re-Fig.i85, 
ceives   an  impulse  in    a    direction   HZ,   not    passing   through   its 
centre  of  gravity  G,   this   impulse    is   transmitted    entirely    to  the 
centre   of  gravity,    which    moves   in    a    direction    parallel   to  HZ 
according  to  which  the  body  has  received  the  impulse ;  and  that 

the  parts  of  this  body,  in  the  mean  time,  turn  about  the  centre  of 
gravity  in  the  same  manner  as  if  it  were  fixed.  Therefore,  if  the 
figure  of  this  body,  and  the  forces  impressed  upon  it  (of  which 
I  suppose  Q  to  represent  the  resultant)  are  such  that  it  can 
turn  only  about  a  single  axis ;  as  this  axis  will  necessarily  pass 
through  the  centre  of  gravity,  all  that  we  have  said  on  the 
subject  of  the  moment  of  inertia,  is  applicable  to  this  case, 
understanding  by  r  in  m  r^,  the  distance  of  any  particle  from  354,  &c. 
the  axis  which  passes  through  the  centre  of  gravity,  and  by  q  X  D 
the  moment  of  the  force  HZ,  taken  with  respect  to  the  same  axis, 
or  the  sum  of  the  moments  of  all  the  forces  which  act  upon  the 
body,  taken  with  respect  to  this  same  axis.  That  is,  the  centre 
of  gravity  will  move  parallel  ^  to  the  direction  of  the  force  g,  with  a 

velocity  =    -|-,   L  being  the  mass  of  the  body  j  and  if  we  draw 

GZ  perpendicular  to  HZ,  and  call  v  the  velocity  of  rotation  of  the     28. 
point  Z,  we  shall  have 

Q  X  GZ         f^„ 
"^  =     fm  r^  •  ^    ^^'  362. 

or 

Q  X  GZ 

J    m  r^ 

Of  this  we  shall  give  a  few  applications. 

380.  Let   us   suppose    that    the    body    JV  impinges  upon    the 

body  L,  according  to  any  direction  whatever  £Q,  in  such  a  man-Fig.i86. 
Mech.  33 


258  Dynamics. 

ner  as  to  cause  no  rotation  in  L,  except  about  a  single  axis  per- 
pendicular to  the  jilane  vvliich  passes  throiigli  the  centre  of 
gravity  G,  and  the  perpendicular  TZ  belonging  to  the  point  of 
>  contact  T;  it  is  proposed  to  determine  the  velocities  after  collision, 
'  •  and  tlie  directions  of  these  velocities,  the  body  L  being  supposed 
at  rest. 

Let  us  imagine  a  plane  touching  the  point  T,  and  let  the  ve- 
locity of  JNT,  according  to  EQ^  be  decomposed  into  two  others 
one  according  to  ET  perpendicular  to  this  plane,  and  the  other 
according  to -E/ parallel  to  this  same  plane.  If  N  had  no  other 
velocity  but  EI,  it  would  only  touch  L  in  passing,  and  would 
communicate  to  it  no  motion,  the  effect  of  friction  being  out  of 
the  question.  It  is  therefore  only  in  virtue  of  the  velocity  ET, 
that  the  impulse  is  produced.  Now  as  it  is  easy  to  determine 
ET  in  the  parallelogram  ETAl,  of  which  all  the  angles  and  the 
diagonal  EA  are  supposed  to  be  known,  we  shall  consider  this 
velocity  £T  as  known  and  we  shall  call  it  u.  Let  u'  represent 
the  velocity  of  N  after  collision,  according  to  the  direction  ET 
or  CZ ;  consequently  u  —  u'  is  the  velocity  lost  by  collision,  and 
N  X  {u  —  u')  is  the  force  impressed  upon  the  body  L,  which 
we  have  called  g.  Therefore  the  centre  of  gravity  and  all  the 
parts  of  the  body  will  move  in  the  direction  GM  parallel  to  CZ, 
Vi^ith  a  velocity 

Nx  (u  —  W) 
V  =  j^ '-  (i), 

calling  this  velocity  v. 

But,  as  the  force  N  X  {u  —  u')  does  not  pass  through   G,  the 

centre  of  gravity  of  L,   the  body  must  turn    about  G,   as  if  this 

136.     point  were  fixed.     Let  v'  be  the  velocity  of  rotation  of  the   point  Z 

where  GZ,  perpendicular  to  CZ,  meets  the  latter  line  j  we  shall 

have,  therefore, 

/  _    Nx{u  —  u')xGZ 


363 


or  representing  GZ  by  D, 


v  = 


/wr2 


(n). 


Rotation  of  Bodies  unconfined.  259 

It  may  be  observed,  moreover,  that  it  is  necessary,  in  order  that 
the  body  JV  may  really  have  the  velocity  u\  that  the  point  T  of 
the  body  L  should  also  have  this  same  velocity  u'  according  to 
TZ.  Let  us  now  see  with  what  velocity  this  point  must  advance 
according  to  TZ. 

It  will  have,  in  the  first  place,  the  velocity  v  common  to  all 
the  parts  of  L.  Moreover,  if  we  suppose  that  the  infinitely  small 
arc  TT'  perpendicular  to  GT,  represents  the  velocity  of  rotation 
of  the  point  T,  by  constructing  the  parallelogram  TCT'B  upon 
the  directions  TT,  TA  and  TZ,  we  shall  have  TZ  for  the  velo- 
city of  T  according  to  TZ  in  virtue  of  its  rotation.  Now  the  sim- 
ilar triangles  TT'C,  GTZ,  give 

GT  :  GZ  ::  TV  :  TC  =   ^^^J'^'. 

But  since  v'  is  the  velocity  of  rotation  of  the  point  Z,  we  have 
v'  :  TT  ::  GZ  :  GT, 

whence 

rprp,   _   V'   X    GT 

^    ~"       GZ      ' 

and  consequently 

therefore  the  total  velocity  of  the  point  T  belonging  to  the  body  £/, 
according  to  EZ,  \s  v  -}-  v' ;  and  hence  v  -{-  v'  =z  u'  (m). 

If  from  the  three  equations  found  above,  in  order  to  express 
the  conditions  of  the  motion,  we  deduce  the  values  of  u',  v',  and  v, 
we  shall  obtain 

,  _       N  (fmr^  +  LD^)  u 


V'    = 


N  uf  m  r^ 
{N -\- L)  J  m  r^  +  LD^  N" 

LND^u 

(iV+  L)  f  m  r'^ -\-  LD^  rr 


260  Dynamics, 

If  the  distance  GZ  or  D  =  0 ;  that  is,  if  the  direction  of  the 

impulse  passes   through  the  centre  of  gravity   G,  the  velocity  of 

rotation  v'  =  0,  the  velocities  u'  and  v  are   equal  to  each  other 

N  u 
and  to  -  ,  as  indeed  they  ought  to  be,  according  to  article 

288.  The  velocity  u'  being  determined,  if  it  be  compounded  with 
the  velocity  El,  which  has  suffered  no  alteration,  we  shall  have  the 
absolute  velocity  of  JV,  and  its  direction  after  collision. 

If  the  body  L  were  in  motion  before  collision,  we  should  de- 
compose the  velocity  of  JV  before  collision  into  two  others,  one  of 
which  should  be  equal  and  parallel  to  that  of  L ;  this  would  con- 
tribute nothing  to  the  impulse,  and  we  should  employ  the  second 
as  we  have  employed  the  velocity  according  to  £Q,  considering 
the  body  L  as  at  rest. 

If  we  compare  the  value  found  above  for  v'  with  that  which  we 
^3-  before  found  for  the  velocity  of  rotation,  by  attending  to  the  differ- 
ence in  the  import  of  r  in  the  two  cases,  we  shall  be  able  to  deter- 
mine the  difference  between  the  velocity  of  rotation  which  belongs 
to  a  free  body,  and  that  belonging  to  one  which  admits  only  of  a 
rotation  about  a  determinate  point  or  axis. 

381.  From  the  value  which  we  have  found  for  v',  the  velocity 
of  rotation,  may  be  deduced  a  method  for  determining  by  experi- 
ment the  value  of /*m  r^,  and  the  position  of  the  centre  of  gravity 
in  a  body  of  any  figure  whatever.  We  shall  apply  to  a  vessel  what 
we  have  to  say  upon  this  subject. 

Let  us  suppose,  that,  by  means  of  a  weight  JV  and  a  rope  at- 
tached near  the  stern,  the  vessel  is  drawn  in  a  direction  perpendic- 
ular to  its  length,  the  weight  being  small  compared  with  the  whole 
weight  of  the  vessel.  Let  this  weight  pass,  for  Instance,  over  the 
Fig.187.  pulley  P.  The  velocity  of  the  vessel  during  the  experiment 
(which  should  continue  only  for  a  very  short  time,  as  a  minute  or 
half  a  minute,)  will  be  so  small  as  to  make  it  unnecessary  to  take  ac- 
count of  the  resistance  of  the  water. 

The  action  of  gravity  communicates  to  JV,  in  the  instant  d  t 

273.    the  velocity  g  d  t  (^g  being  the  velocity  acquired   in  a  second  of 

time),  and  produces  in  the  vessel  an  infinitely  small  velocity  of 

rotation,  which  I  shall  call  d  v'  for  the  point  ^  where  the  rope  is 


Rotation  of  Bodies  unconjined.  261 

attached.  Putting,  therefore,  g  d  t  foT  u,  and  d  v'  for  v',  in  the 
value  of  v'  found  above,  JV  being  considered  as  very  small  or 
nothing  compared  with  L,  the  mass  of  the  vessel,  which  gives 

ND^  u 

J  m  r'^ 

we  shall  have 

,   ,       gND^dt 

dV    =  ^—j: 2—. 

Let  d  v"  be  the  velocity  with  which  that  point  of  the  vessel  turns 
which  is  distant  one  foot  from  the  centre  of  gravity  ;  we  shall  have 

dv'  :  dn"  ::  AG  '.  \  •.'.  B  \  \, 

and  consequently 

dv'  —  B  d  v". 
Substituting  for  d  v'  this  value,  we  have 

J  m  r-' 


and,  by  integrating, 


//      g  ND  t 

v"  = . 


Let  z  be  the  arc  described  by  the  point  in  question  during  the    280. 
time  t ;  we  shall  have 

d  z  =v"dt, 

and  consequently 

J  ^  _  g  ^D  *  ^  ^ . 

f  m  r^ 

whence,  by  integrating, 

gNDf 
'2,  f  rar 


z  =. 


.9  • 


Therefore,  if  the  rope,  acting  always  perpendicularly  to  the 
length  of  the  vessel,  be  attached  to  another  point  /,  and  we  call 
z'  the  arc  described  by  the  same  point  during  the  same  time  t, 
we  shall  have 


262  Dynamics. 

~   ~  2/ffi  r2' 
calling  D'  the  distance  IG ;  whence  we  have 

z  '.  z'  .:  D  '.  D'  ::  AG  '.  IG', 

Alg.224.  ^"d  hence 

:  :  AG  — IG  or  AI  :  AG. 


^  ^^  .*. 


Now  if  after  each  experiment  we  measure,  as  may  easily  be 
done  in  several  ways,  the  angles  of  rotation,  that  is,  the  number 
of  degrees  contained  in  the  arcs  z,  z'  respectively,  we  may  sub- 
stitute these  numbers  instead  of  the  arcs  r,  z'^  in  the  proportion ; 
and  since  the  distance  Al  is  known,  we  readily  obtain  AG^  that  is, 
the  position  of  the  centre  of  gravity. 

Geom.  The  value   of  .4 G  or  D  being  determined,  we  calculate  the 

length  of  the  arc  z  which  has  1  for  radius,  and  of  which  the 
number  of  degrees  is  known  ;  then,  since  N  is  known,  and  g  is 
271.  equal  to  32,2  feet ;  if  we  take  care  to  observe  the  number  of 
seconds  which  elapse  up  to  the  instant  at  which  the  number  of 
degrees  in  z  is  counted,  we  shall  know  every  thing  except  fm.  r^ 
in  the  equation 


_g  ND  t^ 


but  this  equation  gives 


whence  we  obtain  the  value  of  T  w  r^,  which  it  would  be  very 
troublesome  to  obtain  by  a  particular  calculation  of  the  different 
parts  of  the  vessel. 

Fig.188.  382.  When  a  body  L  of  any  figure  whatever,  having  received 
an  impulse  in  a  direction  HZ,  not  passing  through  the  centre  of 
gravity,  takes  the  two  motions  of  which  we  have  spoken,  it  is 
easy  to  see,  that  for  an  instant  it  may  be  regarded  as  having  but 
one  single  motion,  namely,  a  motion  of  rotation  about  a  fixed 
point  or  axis  F,  which  according  to  the  figure  of  the  body,  and 
also  the  distance  GZ  at  which  the  impulse  passes  from  G,  may 
be  situated  either  within  or  without  the  body.     For  if,  while  the 


Rotation  of  Bodies  unconfined.  263 

line  GZ  is  carried  parallel  to  itself  from  GZ  to  G'Z',  we  imagine 
it  to  turn  about  the  movable  point  G,  since  the  points  of  the 
body  have  velocities  of  rotation  greater  in  proportion  to  their 
distances  from  G,  it  is  manifest  that  there  is  upon  the  line  ZG  a 
point  F  which  will  be  found  to  have  described  from  F  toward 
F,  an  arc  equal  to  GG',  and  which  may  be  regarded  for  an  in- 
stant as  a  straight  line  ;  the  point  F  then  will  have  retrograded 
as  far  by  its  motion  of  rotation  as  it  has  advanced  by  the  veloci- 
ty common  to  all  parts  of  the  body ;  this  point  will  therefore 
have  remained  constantly  in  jP,  which,  for  this  reason,  may  be 
considered  for  an  instant,  as  a  fixed  point  about  which  the  body 
turns.  If  we  would  know  the  position  of  the  point  F,  it  will  be 
remarked  that  the  arcs  FF,  Z'l,  which  the  points  F  and  Z  de- 
scribe in  an  instant,  may  be  considered  as  straight  lines  perpen- 
dicular to  GZ,  or  parallel  to  GG';  now  the  similar  triangles 
FFG',  G'ZI,  give 

GZ  :  G'F  ::  ZI  :  FF, 


or 


GZ   :  GF    ::  ZI  :  GG' ; 

but  we  have  found  the  velocity, 


hence. 


therefore 


GG'  =  ^,  and  the  velocity  ZI  =  ^^  ^% 
Li  •'  y  m  r^ 


GZov  D  :  GF  ::  ^^)^   :   | ; 
J  mr^         Li 


GF=   ^" 


383.  The  point  F  is  called  the  centre  of  spontaneous  rotation, 
because  it  is  a  centre  which  the  body  takes  as  it  were  of  itself. 
This  point  is  precisely  the  centre  of  oscillation  which  the  body  L 
would  have,  if  it  turned  about  a  fixed  point  or  axis  situated  in  Z; 
for  from 


GF=  -^"'^' 
"^^  -  Dx  V 


we  have 


264  Dynamics. 


~  GZ  X  L 


Now  Cm  r^  •{■  L  X  GZ  is  m  article  360  precisely  what  we  have 
understood  by  J*m  r^  in  article  361  ;  therefore  the  point  jPis  here 
the  same  as  the  point  O  in  article  361. 

We  perceive,  therefore,  that  the  point  about  which  a  body- 
may  be  considered  as  turning  for  an  instant,  is  independent  of 
the  value  of  the  force  or  forces  which  are  applied  to  this  body  j 
and  generally  it  may  be  inferred  from  the  value  of  FG,  that  this 
point  is  the  more  distant,  according  as  the  force  in  question,  or  the 
resultant  of  all  the  forces,  acts  at  a  less  distance  from  the  centre 
of  gravity. 

361.  384.  We  have   seen  that  when  a  body  turns  about  a  fixed 

point  or  axis,  its  centre  of  percussion  is  the  same  as  its  centre  of 
oscillation ;  whence  these  two  centres  are  found  by  the  same  op- 
eration. It  is  not  the  same  when  the  body  is  free.  For,  let  us 
suppose  a  body  whose  mass  is  L,  to  turn  about  its  centre  of  grav- 
ity with  a  velocity,  which,  for  a  point  situated  at  the  known  dis- 
tance a,  shall  be  v  ;  and  that  at  the  same  time  this  centre  moves 
with  the  velocity  u.  It  is  manifest,  in  the  first  place,  that  the  re- 
sulting  force  of   all  the  motions  belonging  to  the   different  parts 

136.  of  this  body,  will  have  for  its  value  L  X  u  ov  L  u,  that  is,  the 
same  as  if  the  body  had  no  motion  of  rotation.  In  the  second 
place,  the  distance  at  which  the  resultant  must  pass  from  the 
centre  of  gravity,  is  evidently  that  at  which  a  force  equal  to  L  u 
would  produce  in  the  body   a  velocity  of   rotation  equal   to  that 

356,     which  it  actually  has ;  but  this  velocity  v  has  for   its  expression 

J — 2 >  calling  D  the  distance  sought ;  we  have,  therefore, 

L  u  D  a 

^  =  ~r a"' 

J  m  r'' 

and  consequently 


Measure  of  Forces  applied  to  Machines.  265 

_        V         fmr^ 

B  =  -  X  Vt-5 
u  Li  a 

and  hence  we  see  that  the  distance  of  the  centre  of  percussion 
of  a  free  body  depends  on  the  ratio  of  the  velocity  of  rotation  to 
the  velocity  of  the  centre  of  gravity  ;  and  particularly  that  it  is 
nothing  when  the  velocity  of  rotation  is  nothing,  as  in  fact  it  ought 
to  be. 

We  may  hence  determine  at  what  point  to  place  an  obstacle 
in  order  to  stop  a  free  body  which  has  a  progressive  and  rotato- 
ry motion  at  the  same  time ;  namely,  at  the  centre  of  percussion 
of  this  body,  or  the  point  where  it  would  give  the  strongest  blow 
or  exert  the  greatest  force. 


Method  of  estimating  the  Forces  applied  to  Machines. 

385.  Any  force  has  for  its  measure,  as  we  have  already  said, 
the  product  of  a  determinate  mass,  into  the  velocity  which  the 
force  in  question  is  capable  of  giving  to  this  mass.  It  seems 
proper,  in  this  place,  to  add  something  by  way  of  illustrating  the 
application  of  this  principle  to  machines. 

When  two  weights  act  against  each  other  by  means  of  a  sim- 
ple fixed  pulley,  it  is  necessary  in  order  to  an  equilibrium  that 
their  masses  should  be  equal ;  and  this  equilibrium  once  estab- 
lished, will  always  remain. 

But  if  instead  of  opposing  a  weight  to  a  weight,  we  oppose 
the  force  of  an  animal,  as  that  of  a  man,  for  example,  although 
it  be  true  that,  in  order  to  an  equilibrium,  this  man  has  only  to 
exert  an  effort  equal  to  the  weight  to  be  sustained,  that  is,  equal 
to  the  quantity  of  motion  represented  by  the  mass  of  this  body 
multiplied  into  the  velocity  which  gravity  communicates  in  an 
instant ;  it  is,  nevertheless,  evident  that  if  the  man  were  capable 
of  but  one  such  effort,  the  equilibrium  would  continue  only  for 
an  instant,  because  gravity  renews  each  successive  instant  the 
action  which  was  destroyed  in  the  preceding. 

Mech.  34 


272 


266  Dynamics. 

It  is  not,  therefore,  by  the  mass  only  which  the  man  supports, 
that  we  are  to  estimate  his  strength  ;  but  we  must  consider,  also, 
the  number  of  times  that  he  is  able  to  exert  an  action  equal  to 
that  which  gravity  communicates  every  instant  to  the  body.  Now 
if  g  represents  the  velocity  which  gravity  is  capable  of  giving 
to  a  free  body  in  a  second  of  time  ;  and  d  t  represents  an  in- 
267.  finitely  small  portion  of  any  time  t,  g  d  t  will  be  the  velocity 
which  gravity  gives  during  the  instant  d  t,  t  being  supposed  to 
be  reckoned  in  seconds.  Therefore,  if  m  be  the  mass  which  it  is 
proposed  to  sustain,  m  g  d  t  will  be  its  weight,  or  the  quantity  of 
motion  which  gravity  gives  it  each  instant  d  t ;  it  is  accordingly 
the  effort  also  which  must  be  exerted  each  instant  by  the  force 
which  is  to  support  »?,  either  directly  or  by  the  aid  of  a  pulley. 
Therefore,  during  any  time  t,  this  force  must  expend  a  quantity  of 
motion  equal  to 

f  m  g  d  t    or    m  g  t. 

Therefore,  if  i  denotes  the  time  at  the  end  of  w-hich  the  agent  is 
no  longer  able  to  support  the  mass  m,  m  ^  ^  may  be  regarded  as 
the  measure  of  his  strength.  We  do  not  mean  by  this  that  he  is 
no  longer  capable  of  exerting  any  effort ;  but  his  force  having 
become  unequal  to  the  effect  to  be  produced,  it  is  considered  as 
nothing  with  respect  to  this  effect.  Let  us,  for  example,  suppose 
that  in  order  to  support  a  weight  of  50'''-  for  an  hour,  it  is  pro- 
posed to  employ  a  force,  which  acting  by  equal  and  infinitely 
small  degrees,  is  known  to  produce  in  a  mass  of  201''-,  a  velocity 
of  50  feet  in  a  second,  at  the  instant  when  this  force  is  exhausted. 
It  is  manifest  that  this  mass  of  20"'-  will  have  a  quantity  of  motion 
equal  to 

20"'-  X  50  or  1000. 

Let  us  see,  then,  if  this  quantity  of  motion  be  equal  to  what  the 
quantity  m  g  t  becomes,  by  putting  SO"*-  for  m,  an  hour  or  3600'' 
273.  for  t^  and  32,2  feet  for  g.  It  appears  to  fall  far  short  of  it ;  such  a 
force,  therefore,  would  not  support  a  weight  of  50"^-  during  an 
hour.  If  we  wished  to  know  during  what  time,  or  what  number 
of  seconds,  it  would  support  it,  we  have  only  to  suppose 

mg  t  =  1000  ; 


Measure  of  Forces  applied  to  Machines.  267 

and,  putting  50  for  m,  and  32,2  for  g,  we  shall  have 

mff  t  1000  1000        100        5" 

or  t  =.  ^r;: ^rrr-pi  =  TT^TT^  ^=  TFT  ^=  s"  nearly ; 

7n  g  50  X  3'2,2        IblO        Ibl         8  -^  ^ 

that  is,  such  a  force  would  support  a  weight  of  50^''-  only  about  | 
of  a  second. 

386.  Let  us  now  suppose  that  it  is  required  not  only  to  support 
the  mass  m  during  the  time  t,  but  also  to  move  it  during  the  same 
time  with  a  uniform  and  known  velocity  v. 

It  is  manifest  that  in  communicating  the  velocity  v,  either  suc- 
cessively or  at  once,  to  the  body  m,  there  must  have  been  expend- 
ed a  quantity  of  motion  equal  to  jn  v;  and  to  maintain  this  velocity 
V  during  the  time  t,  the  action  of  gravity  is  to  be  resisted  all  the 
while  just  as  if  the  body  had  remained  at  rest ;  that  is,  there  must 
have  been  expended  an  additional  quantity  of  motion  equal  to 
rn  g  t;  therefore  to  maintain  in  the  mass  tn  the  velocity  v  during 
the  time  t,  the  agent  must  be  capable  of  producing  a  quantity  of 
motion  equal  io  m  v  -^  m  g  t. 

387.  It  is  ascertained  by  actual  trial,  that  a  man  can  work  at 
a  machine  like  that  represented  in  figure  97,  for  8  hours  successive- 
ly, and  cause  the  winch  to  make  30  turns  a  minute,  the  radius  of 
the  cylinder  and  that  of  the  winch  being  each  14  inches,  and  the 
weight  applied  at  the  surface  of  the  cylinder  being  25"'-.  This  ex- 
periment determines  the  value  of 

m  V  -{-  m  g  t, 

and  consequently  the  limit  to  be  observed  in  estimating  the  force 
of  a  man  working  at  a  machine,  and  for  a  definite  period  of  time. 
Indeed,  since  the  radius  of  the  winch  and  that  of  the  cylinder  are 
equal,  the  weight  in  this  case  passes  through  the  same  space  with 
the  power.  Thus,  the  radius  being  14  inches,  at  each  turn  the 
power  passes  through  28  X  3,1416,  or  88  inches  nearly;  andccon 
since  it  makes  30  turns  a  minute,  it  describes  44  inches  a  second,  "^^" 
or  II  of  a  foot ;  that  is,  the  velocity 


44        11 

'  =  Vz  =  -2' 


The  mass 


268  Dynamics. 

m  =  25«'-,  g  =  32,2"-,  and  t  =  8^'  =  28800''. 
The  substitutions  being  made,  we  have 

mv  -}-  mgt  =  2|5  ^  23184000  =  23184092. 

By  means  of  this  number,  we  can  judge  whether  the  strength  of  a 
man  be  sufficient  to  produce  a  proposed  effect.  For  instance,  if  it 
be  asked  whether  it  be  possible  for  a  man,  with  the  machine  above 
referred  to,  to  raise  a  weight  of  GO"^-  with  a  velocity  of  10  feet  in  a 
second,  during  0  hours,  we  shall  perceive  that  it  is  not.  For  we 
should  have  in  this  case 

m  =  GO"'- ',  V  =  10',  g  =  32,2 ;  t  =  21600" ; 
which  gives 

mv  -\-  mgt  =  600  -\-  41731200  =  41731800; 

as  this  greatly  exceeds  23184092,  it  follows  that  a  single  man  is 
unequal  to  such  an  effect. 

It  may  be  remarked  that  in  these  two  examples,  the  velocity  v 
with  which  the  man  is  supposed  to  move  the  weight,  is  of  very  little 
consequence  in  estimating  the  force  required  ;  for  in  the  first  exam- 
ple, the  quantity  of  motion  which  answers  to  this  velocity,  is  ^-j*  ; 
and  in  the  second,  600 ;  quantities  which  are  very  small  compared 
with  23IS4092  and  41731800.  Therefore,  in  the  second  exam- 
ple, if  we  are  unable  to  produce  the  desired  effect,  it  is  not  because 
the  velocity  is  greater  than  in  the  first  case,  but  chiefly  because  the 
mass  and  the  time  during  which  it  is  to  be  moved,  require  of  the 
agent  too  great  a  quantity  of  motion. 

While  therefore  the  velocity  required  in  the  agent  is  small  com- 
pared with  g  t,  that  is,  with  the  velocity  which  a  heavy  body  falling 
freely  would  acquire  in  the  time  during  which  the  agent  is  sup- 
posed to  be  employed,  we  may  take  simply  for  the  measure  of  the 
force  in  question,  the  quantity  mgt;  and  we  shall  have 

mgt  =  23184000. 

Thus,  if  the  mass  (the  velocity  with  which  it  is  to  be  moved  be- 
ing moderate)  multiplied  by  the  velocity  which  a  heavy  body  fall- 
ing freely  would   acquire    in   the   time   during   which   the   power 


Measure  of  Forces  applied  to  Machines.  269 

is  to  act,  forms  a  product  less  than  the  constant  number  231S4000, 
or  exceeding  it  but  a  little,  the  power  may  be  considered  as  suf- 
ficient for  the  proposed  effect,  it  being  supposed  to  act  as  in  the 
two  preceding  examples.  But  if  the  velocity  with  which  the 
weight  is  to  be  moved,  is  considerable  compared  with  g  i,  it  will 
be  necessary  to  subtract  from  the  constant  number  23J  84092,  the 
quantity  of  motion  m  v  due  to  the  velocity  with  which  the  body 
is  to  be  moved  ;  and  if  the  weight  multiplied  by  g  t,  the  velocity 
which  a  falling  body  would  acquire  in  the  time  during  which  the 
machine  is  to  be  worked,  forms  a  product  smaller  than  the  remainder 
above  found,  the  power  may  be  deemed  sufficient. 

38S.  In  what  we  have  now  said,  we  have  taken  no  account 
of  friction.  When  the  mouon  of  the  machine  has  become  uni- 
form, (which  is  the  state  in  which  machines  ought  to  be  consider- 
ed) the  effect  of  friction  may  be  regarded  as  constant,  and  it 
may  be  compared  to  a  new  mass  required  to  be  moved  together 
with  the  proposed  mass.     Thus,  in  the  case  above  considered,  the 

friction  being  supposed  equivalent  to  the  weight  of  a  known  part  — 

of  the  mass  m,  this  resistance  will  require  in  the  power  a  quantity  of 
a 
c 


motion  equal  \o  —  m  g  t.,  and  thus. 


m  V  -| m  g  t  -\-  mg  t, 

or 

mv  -}-Q  -\-  l)mgt 

will  be  the  measure  of  the  moving  force. 

If  then,  in  the  experiment  above  referred  to  (the  axle  being 
supposed  to  have  a  radius  much  less  than  that  of  the  cylinder), 
we  suppose  the  friction  to  have  been  y\  of  the  weight,  m  v  being 
neglected,  as  it  may  be  in  this  case,  we  must  augment  the  number 
23184000  by  its  twelfth  part ;  then  the  force  of  a  man  in  similar 
circumstances  may  be  represented  by  the  number  2511G000. 
We  see,  therefore,  that  to  be  able  to  estimate  with  sufficient  ac- 
curacy the  force  of  a  man,  we  must  previously  ascertain  the  ratio  ol 
the  force  of  friction  to  that  of  the  weight,  in  the  experiment 
employed,   with  the  view  of  determining  this  force.      Then  if  fc 


270  Dynamics. 

be  the  value  derived  from  this  experiment  for   (  — \-  1 )  m  g  t, 
we  shall  have, 

neglecting  m  v,  when  v  is  small  compared  with  g  t.      This  equa- 
tion   will    enable    us   to  judge    for   any   other  supposed   value  of 

-,  whether  the  force  of  a  man  will  be  sufHcient  to  move  the  weight 
c 

m  during  the  proposed  time  i. 

389.  In  all  that  we  have  now  said,  we  have  regarded  the 
agent  as  acting  immediately  upon  the  weight,  and  as  deriving 
no  advantage  from  local  circumstances  and  the  nature  of  the 
machine.  We  may  often  rely  upon  a  much  greater  effect  than 
the  particular  considerations  now  presented  would  lead  us  to  ex- 
pect. For  instance,  in  the  use  of  the  pulley  a  man  may  add  to 
his  own  proper  force  the  weight  of  his  body,  or  a  large  part  of  it. 
There  are,  moreover,  many  other  circumstances  of  which  he  may 
avail  himself,  and  other  machines  which  admit  of  similar  expedients. 
Frequently  the  motion  is  not  continued,  but  takes  place  by  starts, 
as  in  the  pulley ;  and  if  there  is  a  loss  on  this  account,  there 
is  also  this  advantage,  that  the  agent  by  intervals  of  rest  is  capable 
of  exerting  the  same  action  for  a  longer  time.  We  shall  not 
dwell  upon  these  details  which  it  will  be  always  easy  to  take  into 
the  account  after  all  that  has  been  said,  especially  if  we  proceed 
according  to  experiments  in  which  care  has  been  taken  to  distinguish 
the  several  causes  on  which  the  action  of  the  moving  force  depends, 
and  to  note  what  belongs  to  each. 

It  is  commonly  said  that  a  mun  can  continue  during  about 
eight  hours,  to  exert  an  effort  equal  to  25"'-.  It  will  be  seen  from 
what  precedes,  that  such  a  statement  does  not  sufficiently  deter- 
mine the  value  of  the  force  in  question  ;  besides,  it  is  necessary, 
as  we  shall  soon  undertake  to  show,  to  have  regard  to  the  velocity 
with  which  the  man  acts ;  it  is  no  less  necessary  to  consider  also 
the  manner  in  which  the  action  is  applied,  and  many  other  circum- 
stances which  we  cannot  now  stop  to  enumerate.  It  is  proper, 
when  circumstances  vary,  to  proceed  in  our  calculations  upon 
new   experiments  made  with  reference   to   these    circumstances. 


Measure  of  Forces  applied  to  Machines.  271 

390.  Although  we  have  considered  that  case  only,  in  which 

the  weight  transmits  all  its  resistance  to  the   power,  it  is  not  less 

easy,  after  what  has  been  said  respecting  the  ratio  of  the  weight 

to  the  power  in   each  machine,  to  determine  whether  by  the  aid 

of  a  particular  machine,  a  given  power  will  produce  a  proposed 

effect.     In  die  wheel  and  axle,  for  instance,  if  the  radius  of  the 

cylinder  be  d,  and  that  of  the  wheel  D ;   in  order  that  the   weight 

may  move    with   the    velocity   v,  it  is  necessary  that  the  power 

11,1  •         [•         ■  1  m  V  8         ... 

should    have   a  quantity  oi    motion   equal   to   — Yr~  ?  ^"^"^  since  in 

the  time  t,  the  action  of  gravity  would  give  to  the  body  m  the 
quantity  of  motion  m  g  t,  the  power  in  order  to  sustain   this  effort 

must  have  the   force  or  quantity  of  motion  —^ — ;  finally,  if  the 

friction  is  equivalent  to  the  -  part  of  the  weight,  m  being  supposed 

to  be  applied  at  the  distance  d,  the  power  will  require  the  addi- 

,  .         -.        .      a       m  ff  t  d       ,         .         ,  , 

tional  quantity  ot  motion  -  X  — jx —  ;  thus,  in  order  to  deter- 
mine whether  the  power  be  sufficient  to  move  with  the  velocity 
V  during  the  time  t,  the  mass  m,  upon  a  wheel  and  axle  of  which 
the  radius  of  the  axle  is  d,  and  that  of  the  wheel  D,  we  must  de- 
termine by  experiment  the  value  of 

m  V  8 


by   employing  at  a    wheel  and  axle,    of  known   dimensions    and 
known  friction,   a  man  moving   a  known  mass ;  then  if  7c  is  the 

value  found  by  putting  for  w,  v,  5,  D,  -,   and    ^,    the    values   of 

these  quantities  respectively  in  the  experiment,  it  will  be  necessary, 
in  every  other  case,  that 

mv  8        /a  \mi^t8 

should  have  a  value  not  exceeding  k. 

So  also,   upon  the  inclined    plane,    the  power  acting  parallel 
to  the  plane  ;  if  we  call  i  the  inclination  of  the  plane,  m  g  t  sm  i   207. 
will    be   the   quantity  of  motion  which   gravity  will  communicate 


272  Dynamics. 

successively  to  the  movable  body,  according  to  the  directions  of 
the  plane,  in  the  time  t ;  thus  the  power  will  be  required  to  have  a 
quantity  of  motion  equal  to 

m  V  -\-  m  g  t  sin  i ', 


and  if  the  friction  be  the  —  part  of  the  weight,  it  will  be  necessary 
to  employ  a  quantity  of  motion  equal  to 


m  V  -\-  m  g  t  sin  i  -j — -  m  g  t. 


Having,  therefore,  determined  by  experiment  one  value  of 


a 


m  V  -{-  m  g  t  sm  I  -\ m  g  t, 


it  will  be  necessary  when  we  wish  to  determine  whether  the 
same  power  be  capable  of  moving  a  given  mass  m,  with  a  known 
velocity  v,  during  a  known  time  t,  upon  a  plane  whose  inclina- 
tion is  i,  and  upon  which  the  friction  is  a  known  part  of  the 
weight ;  it  will  be  necessary,  I  say,  to  determine  whether  the  value 
which 

mv-\-mgtsini-\ —  ^  g  ^ 

will  then  have,  is  less  than  that  in  the  experiment,  or  at  most  only 
equal  to  it ;  in  either  case  the  thing  will  be  possible. 

If  the  time  t,  during  v/hlch  the  machine  is  to  be  in  motion,  be 
not  given  ;  still  if  we  know  the  space  which  the  power  or  the 
weight  must  describe  with  the  velocity  v  ;  then,  as  we  suppose  that 
the  motion  is  uniform,  if  we  call  s  the  space   which  the  weight  is  to 

24.      pass  through,  we  should  put  instead  of  t  its  value  — . 


Such  is,  in  substance,  the  method  which  is  to  be  pursued  in 
estimating  forces  ap|)lied  to  machines.  Each  machine  requires 
particular  considerations  as  to  the  nature  of  the  power  and  the 
manner  in  which  it  is  applied  to  this  machine.  But  by  going 
back  to  the  quantity  of  motion  to  be  expended  by  the  agent,  we 
may    always   determine   whether    he   be   capable   of  a  proposed 


Maximum  Effect  of  Agents.  273 

effect ;  and  the  principles  which  we  have  now  laid  down,  will  serve 
to  conduct  us  in  such  inquiries. 


Of  the  Maximum  Effect  of  Agents  and  Machines. 

391.  When   any  power  is  made  to  act  upon  a  given   resistance, 
by  the  intervention  either  of  a  simple  or  a  compound  machine,   an 
equilibrium  will  take  place  when  the  velocity  of  the   power  is  to 
the  velocity  of  the  resistance  as  the  weight  is  to  the  power.     In 
this   state    of  things,    however,   the   machine   must  be    actually  at    22" 
rest,    and   therefore    incapable    of  performing    any    work.     If  we 
can  increase  the    power,  the  machine  will  move   with  more   and 
more    velocity,    and    will  have    its    motion    gradually    accelerated 
as  long   as  the   power  exceeds   the  resistance.     But   if  from   any 
cause  the  power  should    begin    to    diminish,   or  if  the    resistance 
should   increase,  or  if  both   these  changes  in  the  slate  of  the   ma- 
chine should  take   place  at  the  same  time,  the   acceleration  of  the 
machine  will  diminish,  and   it  will  at  last  arrive  at  a  state  of  uni- 
form   motion.     Now  this    increase    of    resistance    may    arise    in 
many  cases   from   an   increase  of  friction,  which  often  (though  not 
always)    accompanies    an    augmentation    of    velocity ;    or   it   may 
arise    from    the    resistance   of  the    air,  which  must   necessarily  in- 
crease  with    the   velocity ;  and    therefore   all    machines  are   found 
soon    to   attain    a  state   of  uniform    motion.     When  an   undershot 
wheel  is    driven  by  the  impulse  of  water,  the  uniformity  of  motion 
to  which    it  arrives,  arises   principally  from    the   diminution  of  the 
power    which    in    this   case  accompanies    an    increase  of  velocity. 
When  the  mass  of  fluid    strikes  one  of  the  float-boards  at  rest,  the 
impulse   is  then  a  maximum.     When   the  float-board  is  in   motion 
it  withdraws  itself,  as  it  were,  from   the  action  of  the   power,   and 
therefore    its    mechanical  effect  will    diminish    as    the    velocity    in- 
creases,   and    if  it   were    possible   that   the   velocity  of  the  wheel 
should    become    equal    to  that  of  the  fluid,  the  float-board   would 
not  be  struck  at    all  by  the  moving  water.     Hence  it  follows,  that 
the  power  itself  diminishes  by  an  increase  of  velocity,  and  there- 
fore that  from    this    cause  alone   machines  in  general  would   soon 
acquire    a    motion    sensibly    uniform.      This    effect  will    be    more 
easily  understood,  if  we  suppose  an   axle   to  be   put  in    motion   by 

Mcch.  35 


274  Dynamics. 

two  currents  of  water,  moving  with  different  velocities  and  driv- 
ing two  wheels,  one  of  which  is  placed  at  each  extremity  of  the 
axle.  When  the  wheels  have  begun  to  move,  by  the  joint  action 
of  these  falls  of  water,  its  motion  will  at  first  be  slow,  and  each 
fall  of  water  will  jDcrform  its  part  in  giving  motion  to  the  axle ; 
but  if  the  greater  fall  is  capable,  by  the  continuance  of  its  action, 
of  giving  its  wheel  a  velocity  either  equal  to,  or  greater  than  the 
velocity  of  the  smaller  fall,  then  it  is  manifest  that  the  smaller 
fall  ceases  to  impel  its  wheel,  and  that  the  whole  effect  is  produced 
by  the  action  of  the  greater  fall.  Hence  it  will  be  perceived  from 
this  statement,  not  only  why  a  diminution  of  the  impelling  power 
accompanies  an  increase  of  velocity,  but  why  there  is  a  certain 
velocity  of  the  machine,  which  is  necessary  before  we  can  gain 
all  the  useful  effect  which  we  wish  to  have  from  the  powers  which 
we  employ. 

392.  In  order  to  illustrate  this  in  the  case  of  a  real  machine, 
let  us  suppose  that  the  power  of  a  man  is  to  be  employed  in  rais- 
ing a  load  by  means  of  a  walking  crane.  This  machine  consists 
of  a  large  wheel  placed  upon  an  axle,  round  which  is  coiled  a  rope, 
having  a  weight  r  attached  to  its  lower  extremity ;  the  man 
walks  upon  the  interior  of  the  wheel,  and  by  his  weight  gives 
it  a  rotatory  motion,  and  thereby  coils  the  rope  round  the  axle, 
and  elevates  the  weight  r.  Let  us  suppose  the  wheel  or  drum 
so  constructed,  like  the  fusee  of  a  watch,  that  the  man  can 
walk  at  different  distances  from  the  axis ;  and  let  p  be  the  power 
or  weight  of  the  man,  r  the  weight  to  be  raised,  and  d  the  distance 
of  the  latter  or  radius  of  the  axle,  and  d  the  distance  of  the  former 
or  the  radius  of  the  wheel ;  then 

r  d 

the  distance  from  the  centre  of  the  wheel,  at  which   the  man  must 

place  himself,   in  order  to   be   in   equilibrium  with  the  resistance  r. 

But  as  the  machine  must  be  moved,  and  the  weight  raised,  the  man 

r  8 
must  go  to  a  greater  distance  from  the  axis  than ;  the  motion  of 

P 

the    machine  will    therefore    be   accelerated,    and    the    acceleration 

would  increase  as  he  moved  to  a  greater  and  greater  distance 
from  the  centre  of  the  wheel.  Hence  it  is  obvious,  that  as  the 
acceleration    increases,    the    man    must    walk   with    greater   and 


Maximum  Effect  of  Agents.  275 

greater  velocity ;  but  there  is  an  obvious  limit  to  this,  for  he 
would  soon  be  fatigued  by  the  rapid  walking,  and  would 
therefore  be  rendered  unfit  to  continue  his  work.  He  must 
therefore  return  to  that  distance  from  the  axis,  where  the  wheel 
has  such  a  velocity  that  he  can  continue  to  move  with  that  velocity 
during  the  period  that  his  work  is  to  last  ;  that  is,  there  is 
a  particular  velocity  with  which  the  man  must  walk,  in  order  to 
perform  the  greatest  quantity  of  work ;  and  it  would  be  easy 
to  find  this  velocity,  if  we  knew  the  law  according  to  which  his 
force  is  diminished,  as  his  velocity  increases.  We  may  suppose, 
however,  that  his  force  diminishes  in  the  same  ratio  as  his  velocity 
increases. 

393.  Let  p  represent  the  force  which  a  man  can  exert  during 
a  given  time  against  a  dead  weight.  This  force  will  obviously 
be  greater  than  any  which  he  would  exert  on  the  supposition  of 
motion  taking  place  ;  for  a  part  of  his  strength  in  this  case  would 
be  expended  in  putting  himself  in  motion  and  in  continuing  this 
motion.  Let  v  be  the  velocity  with  which  he  would  lose  the 
power  of  exerting  any  force  :  then,  if  he  move  with  a  velocity  v' 
less  than  v,  he  will  exert  a  force  less  than  p,  and  the  part  lost  may 
be  found,  according  to  the  above  hypothesis,  that  the  diminution  of 
force  is  as  the  increase  of  velocity.  Since  he  loses  all  his  force, 
or  p,  when  the  velocity  is  v,  and  none  when  there  is  no  velocity,  we 
have 

p  V 
V  —  0  or  u  :  v'  —  0  or  v'  :  :  p  :  - — • 

I 
- —  is  therefore  the  loss  of  force  sustained  on  account  of  moving 

V 

with  the  velocity  ■«'.     There  will  accordingly  remain 

pv'  /,  v'\ 

as  the  effective  force  actually  exerted  against  the  weight.  Now  if 
D  be  the  distance  at  which  this  force  acts,  r  the  resistance  or  weight 
raised,  and  8  the  distance  at  which  the  resistance  acts,  and  u  its 
velocity  ;  then,  when  the  machine  has  attained  a  uniform  motion, 
we  shall  have 


(■-.)■> 


:=  r  8. 


276  Dynamics. 

But,  since 

D  :  d  :  :  v'  :  u, 

we  have,  by  substitution, 

p  (  i  —  —  j  v'  =  r  u. 

To  find  the  maximum  we  put  tiie  differential  of  the  first  member 
equal  to  zero,  v'  being  regarded  as  variable ;  we  have  thus 

Cal.  45.  pdv' ^ =  0, 

or 

t)  =  2  u'     and     v'  =z  ^  v. 

Substituting  ^  v  for  v'  in  the  above  equation,  we  obtain 

r  u  =:  I  p  V. 

On  the  hypothesis,  therefore,  which  we  have  assumed,  the  man  will 
do  most  work  when  he  moves  with  half  his  greatest  velocity,  and  in 
this  case  the  greatest  effect  will  he  I  p  v. 

394.  It  appears,  however,  by  direct  experiments,  that  the  force 
of  a  man  diminishes  as  tiie  square  of  his  velocity  increases,*  in 
other  words,  that  the  effective  forces  are  as  the  squares  of  the 
diminutions  of  velocity  from  the  point  where  the  effective  force  is 
nothing.  Calling  p',  therefore,  the  force  answering  to  the  velocity 
i/j  we  shall  have,  according  to  this  hypothesis, 

p  '.  p'  \  :  {v  —  0)^  :  {y  —  v'f ; 
whence 

'P'  =V  \—^^    =P  (^)    CO'  P"«'"S  v  —  v'  =  w  (ii) ; 
and  hence, 

P  (j,)    ^'  or  i^  (j;)    (^  —  ^^')  =  ^  ^h 
since 

Taking  the  differential,  as  before,  and  putting  it  equal  to  zero,  and 
suppressing  the  constant  factor,  we  have 

*  See  note  on  the  measure  of  forces. 


Maximum  Effect  of  Machines.  277 

2  V  w  d  w  —  2  iv^  d  w  =:  0  ; 

whence  2v  =  Sw,  and  w  =:  ^  v.    Substituting  this  value  in  equa- 
tions (ii),  (i),  we  obtain 

V    v'  :=   ^  V, 

or 


also 


p' =pQiy= ip'^ 


that  is,  the  work  done  is  a  maximum  when  the  agent  moves  with  one 
third  part  of  the  greatest  velocity  of  which  he  is  capable,  and 
when  the  weight  or  load  is  f  of  the  greatest  which  he  is  able  to  put 
in  motion  during  the  whole  time  he  is  supposed  to  act. 

395.  Having  thus  considered  the  maximum  effect  of  living 
agents,  we  shall  proceed  to  the  subject  of  machines,  and  shall  take 
the  case  of  a  wheel  and  axle,  as  almost  all  other  machines  may  be 
reduced  to  this. 

Tlie  powers  by  which  a  machine  is  put  in  motion,  and  by  which 
that  motion  is  kept  up,  are  called  first  movers,  or  moving  powers, 
or  more  familiarly,  mechanical  agents ;  and  when  various  moving 
powers  are  applied  to  the  same  machine,  the  resultant  of  them, 
or  the  equivalent  force,   is  called  the  moving  force. 

The  first  movers  of  machinery,  are,  the  force  of  men  and  that 
of  other  animals,  the  force  of  steam,  the  force  of  wind,  the  force 
of  moving  water,  the  weight  of  water,  the  reaction  of  water,  the 
descent  of  a  weight,  the  elasticity  of  a  spring,  &;c.  If  a  machine 
be  driven  by  two  powers  acting  in  two  different  directions,  we 
must  then  find  their  resultant,  and  consider  the  machine  as  driven 
by  the  resulting  force. 

The  powers  which  oppose  the  production  of  motion  in  a  ma- 
chine, and  its  continuance,  are  called  resistances  ;  and  the  resultant 
of  all  the  resisting  forces  is  called  the  risistance. 

The  work  to  be  performed  is,  in  general,  the  principal  resist- 
ance to  be  overcome  ;  but,  in  addition  to  this,  we   must  consider 


278  Dynamics. 

the  resistance  of  friction,  and  the  resistance  arising  from  the  in- 
ertia of  all  the  parts  of  the  machinery  ;  for  a  certain  portion  of 
the  moving  power  is  necessarily  wasted  in  overcoming  these  obsta- 
cles to  motion. 

The  impelled  point  of  a  machine  is  that  point  at  which  the 
moving  power  is  applied,  or  rather  that  point  at  which  the  mov- 
ing force  is  supposed  to  act,  when  this  moving  force  is  the  result- 
ant of  various  powers  differently  applied.  The  working  point  of 
a  machine  is  that  point  at  which  the  resistance  is  overcome,  or 
that  point  at  which  the  resultant  of  all  the  resisting  forces  is  sup- 
posed to  act. 

The  work  performed,  or  the  effect  of  a  machine,  is  equal  to  the 
resistance  multiplied  by  the  velocity  of  the  working  point. 

The  moment  of  impulse  is  equal  to  the  moving  force  multiplied 
by  the  velocity  of  the  impelled  point. 

396.  In  proceeding  to  investigate  general  expressions  for  the 
ratio  of  the  velocities  of  the  impelled  and  working  points  of  ma- 
chines, when  their  performance  is  a  maximum,  let 

D  =  the  radius  of  the  wheel  to  which  the  power  is  applied  ;  or, 
which  is  the  same  thing,  the  velocity  of  the  impelled  point 
of  the  machine ; 

8  =  the  radius  of  the  axle  to  which  the  resistance  is  applied,  or 
the  velocity  of  the  working  point  of  the  machine  ; 

p  =z  the  moving  force  applied  at  the  impelled  point ; 

r  =  the  resistance  arising  solely  from  the  work  to  be  per- 
formed ; 

m  =■  the  inertia  of  the  moving  power  p,  or  the  quantity  of  matter 
to  which  that  power  must  communicate  the  velocity  of  the 
impelled  point  j 

n  =  the  inertja  of  the  resistance,  or  the  quantity  of  matter  to  be 
moved  with  the  velocity  of  the  working  point,  before  any 
work  can  be  performed  ; 

/  :=  the  quantity  of  matter,  which,  if  placed  at  the  working 
point,  would  create  the  same  resistance  as  friction  ; 


Maximum  Effect  of  Machines.  279 

i  =  the  quantity  of  matter,  which  if  placed  at  the  working  point, 
would  oppose  the  same  resistance  as  the  inertia  of  all  the 
parts  of  the  machinery. 

Since  d  and  S  are   the  radii  of  the  wheel  and  axle,   we  shall 
have   B  :  d  : :  r  :  — ,  a  weight  equal  to  that  part  of  the  power  p 

which  is  in  equilibrium  with  the  resistance.     We   have,  therefore, 

f  8  .  .  • 

p as  an  expression  for  the  effective  force  of  the   power  ;  and 

as  D  is  the  distance  at  which  this  force  is  applied,  we  have 

p  D  —  r  8 

to  represent  the  force  which  is  employed  in  giving  a  rotatory 
motion  to  the  machine.  The  resistance  which  friction  opposes 
to  this  force  will  hef  8',  the  moment  of  inertia  of  the  power  p 
will  be  as  m  d^  ;  the  moment  of  inertia  of  the  resistance  as  n  8% 
and  the  moment  of  inertia  of  the  machinery  will  be  as  i  8^. 
Since  the  moving  force  is  diminished  by  the  resistance  of  friction, 
we  shall  have  po  —  r  8 — f8  for  the  moving  force  ;  and  since 
the  resistance  arises  from  the  moment  of  inertia  of  the  resistance, 
the  moment  of  Inertia  of  the  power,  and  that  of  the  machinery,  it 
will  be  as  TO  D"  -}-  n  8^  -\-  i  8^.  But  the  velocity  Is  propor- 
tional to  the  moving  force  directly  and  to  the  resistance  inversely ; 
therefore 


the  rotatory  velocity  will  be 


p  D  —  r  8  — f  8 

m  d'2  -\-  71  8^  -\-  i  8^' 

Now,  since  the  velocities  of  the  impelled  and  working  points  are  as 
their  distances  from  the  centre  of  motion,  or  as  d  and  8,  we  shall 
obtain  these  velocities  respectively  by  multiplying  the  rotatory 
velocity  by  d  and  8 ;  and  as  the  work  performed  is  equal  to 
the    resistance    multiplied    by    the  velocity  of  the  working  point ; 

we  shall  have  for  the  velocity  of  the  impelled  point 

p  1)2  —  r  v  8 — fv8^ 
m  d2  4-  n  (52  -f-  i  8^'  ' 

for  the  velocity  of  the  working  point 


280  Dynamics. 

pi>  8  — r  8^-/5^    ^ 
m  d2  +  n  (52  +  i  (52     ' 

and  for  the  work  performed 

r  p  v  8  —  r^  8^  —  rj"  8^ 

m  D^  -\-  n  8^  -i-  i  8^ 

In  order  to  obtain  absolute  measures  of  the  velocities  and  the 
work  performed,  we  must  consider  that,  q  being  the  accelerating 
force,  and  q  g  the  velocity  acquired  in  a  second,  we  shall  have 
1  :  t  :  :  q  g  :  V  =  q  g  i;  and  as  the  accelerating  forces  are  pro- 
portional to  the  velocities  generated  by  them  in  equal  times,  the 
preceding  expressions  for  the  velocities  of  the  impelled  and 
working  points  may  be  substituted  for  the  accelerating  force  q 
in  the  equation  v  =:  q  g  t,  and  we  shall  obtain,  for  the  absolute  ve- 
locity of  the  impelled  point 

p  T>^  —  r  V  8  — f  J)  8 
m  d2  +  n  8^T8^~   ^  S^'j 

for  the  absolute  velocity  of  the  working  point, 

pj,8  —  r8^-—f8'^ 
m  d2  +  n  (52  -1-  i  ,52      ^  ^     ' 

and  for  the  work  performed 

rp  V  8  —  ?"2  52  —  rf  8^    ^ 

This  is  a  maximum  when   the  differential,  8  being  considered   as 
variable,  is  equal  to  zero,  which  gives 

{pjy  —  2  8{r  +/)  )  (m  d2  +  8^  (n  -f-  i)  ) 
—  28{n  +  i){pD8  —  8^r  +/)  )  =  0, 

or,  by  reducing, 

p  m  i>^  —  p  o  8~  (?i  -f-  i)  —  2  (5  m  d^  (r  -|-/)  =  0  ; 

that  is, 

52  + 


2mD{r-\-f)8  m  b^ 


p  {n  -\-  i)  »  -f  i' 


Maximum  Effect  of  Machines.  281 

Resolving  this  after  the  manner  of  an  equation  of  the  second  de- 
gree, we  obtain 


P  C»  +  i) 
When  r  =:  0,  we  have 


^/?«"^  f^  -\-  p^  m  {71  4-  i)  —  ^f 

0  =:  B   ^ 7 j — TT . 

p{n-\-i) 

This  case  takes  place  when  the  resistance  to  be  overcome  exerts  a 
contrary  strain  on  the  machine,  while  it  consists  merely  in  the  iner- 
tia of  the  impelled  body  ;  as  in  driving  a  millstone,  a  fly,  or  in 
pushing  a  body  along  a  horizontal  plane. 

When  /  =  0, 


V'm^  r^  -j-  p2  „j  ^^  _j_  c^  ^^  jii  r 

~  P  ('*  +  i) 

This  case  takes  place  when  the  friction  is  so  small  that  it  may  be 
disregarded,  which  often  happens  in  good  wheel-work,  where  the 
surfaces  that  touch  one  another  are  very  small. 

When  r  =  0,  and  f  =z  0,  we  have 

p  [n  -{-  i)  ^  p^  (n  -j-  z)2  -^n  -j-  « 

This  case  takes  place  when  the  circumstances  of  the  two  preceding 
cases  are  combined. 

When  n  =r  0,  we  have 


8  =  0  v/^»^  (>•  +  /)^  +  P^  "^  ^—  >"  (^  +/) 
pi 

This  case  takes  place  in  the  grinding  of  corn,  the  sawing  of  wood, 
the  boring  of  wooden  or  iron  cylinders,  &ic.,  where  the   quantity  of 
motion  communicated  to  the  flour,  the  saw-dust,  or  the  iron  filings,  is 
too  trifling  to  be  taken  into  the  account. 
Mech.  36 


282  Dynamicis. 

When  r  =  0,  /  =  0,  and  n  =  0,  we  have  5  =  d    y^. 
When  m  :  n  '.  :  p  '.  r^we  have, 


a  =  D 


p  (r  +  z) 


This  case  takes  place  when  the  inertia  of  the  power  and  the  resist- 
ance are  proportional  to  their  pressure  ;  as  when  water,  minerals, 
or  any  other  heavy  body,  is  raised  by  means  of  water  acting  by  its 
weight  in  the  buckets  of  an  overshot  wheel. 

When,  in  the  last  case,  i  =  0,  and  f  =  0,  we  have 


r 


This  case  often  takes  place,  and  particularly  in  pulleys ;  and  making 
D  =  1,  and  r  :=  1,  we  obtain 


S  =  \/p  +  I  —I; 
and  when  p  =z  1,  and  d  =  1,  we  have 


d 


JJ  +  '->- 


The  preceding  formulas  will  be  found  applicable  to  almost  every 
case  which  can  occur ;  and  the  intelligent  engineer  will  have  no 
difficulty  in  accommodating  them  to  any  unforeseen  circumstan- 
ces. 

The  following  table  will,  in  many  cases,  save  the  trouble   of 
calculation.     It  is  computed  from  the  formula 

D  being  supposed  =  1,  and  r  r=  10. 


Maocimum  Effect  of  Machines. 


283 


Table  containing  the  best  Proportions  between  the  Velocities  of  the 
Impelled  and  Working  Points  of  a  Machine,  or  beiwetn  the 
Levers  by  which  the  Power  and  Resistance  act. 


Proportion- 
al    value     of 
the  impelling 
power,  or 
V- 

Value'  of  the  velocities 
of  the   working  point, 
or  S,  or  of  the  lever,  by 
which    the    resistance 
acts,  that  of    d    being 
1. 

Proportion- 
al    value     of 
the  impelling 
power,  or 
T- 

Value  of  the  velocities 
of  the  working   point, 
or  (5,  or  of  the  lever,  by 
which    the    resistance 
acts,  that  of  d   being 
1. 

1 

0.04S809 

20 

0.732051 

2 

0.095445 

21 

0.760682 

3 

0.140175 

22 

0.788854 

4 

0.183216 

23 

0.816590 

5 

0.224745 

24 

0.843900 

6 

0.264911 

25 

0.870800 

7 

0.303841 

26 

0.897300 

8 

0.341641 

27 

0.923500 

9 

0.378405 

28 

0.949400 

10 

0.414214 

29 

0.974800 

11 

0.449138 

30 

1.000000 

12 

0.483240 

40 

1.236200 

13 

0.516575 

50 

1.449500 

14 

0.549193 

60 

1.645700 

15 

0.5811.39 

70 

1.828400 

16 

0.612451 

80 

2.000000 

17 

0.643168 

90 

2.162300 

18 

0.673320 

100 

2.316600 

19 

0.702938 

In  order  to  understand  the  method  of  using  this  table,  let 
us  suppose  that  we  wish  to  raise  two  cubic  feet  of  water  in  a 
second,  by  means  of  the  power  of  a  stream  which  affords  five' 
cubic  feet  of  water  in  a  second,  applied  to  a  wheel  and  axle, 
the  diameter  of  the  wheel  being  seven  feet.  It  is  required, 
therefore,  to  find  the  diameter  which  we  must  give  to  the  axle, 
in  order  to  obtain  a  maximum  effect.     We  have  obviously  p  =  5, 

5 

and  r  =  2,  and  since  p  :  r  :  :  5  :  2,  we  have  p  z=z  —r;  but,  in 

5 

the  above  table,  r  =:  10  ;  hence  p  =  —r-  10  =  25.     Now  it  ap- 

pears  from  the  table,  that  when  p  :=.  25,  the  diameter  of  the 
axle,  or  d,  is  0.8708,  d  being  1  ;  but  as  d  =  7,  the  diameter  of  the 
axle  must  be  7  X  0.8708  =  6.0956. 


284  Dynamics. 

397.  When  a  machine  is  already  constructed,  the  velocity 
of  its  impelled  and  working  points  are  determined;  and  therefore 
in  order  to  obtain  from  it  its  maximum  effect,  we  must  seek  for 
the  best  proportion  between  the  power  and  the  resistance,  as  tliese 
are  the  only  circumstances  over  which  we  have  any  control,  without 
altering  the  machinery. 

In  order  to  find  the  ratio  of  p  to  r,  which  would  produce  a 
maximum  effect,  it  is  requisite  only  to  make  r  variable  in  the 
formula  above  given  ;  but  it  often  happens,  that  when  r  varies, 
the  mass  ii  suffers  a  considerable  change,  although  there  are  other 
cases  when  the  change  in  n  is  too  inconsiderable  to  be  no- 
ticed. 

Let  us,  therefore,  first  take  the  case  when  r  alone  varies  with- 
out inducing  a  change  in  n.  In  this  case,  the  expression  for  the 
work  performed,  namely, 

r  p  n  8  —  ?-2  52  __  ^y  ^2 
m  d2  +  n  8'^  -}-  i  ,52      ' 

will  be  a  maximum  when 


as  will  be  readily  found  by  differentiating,  putting  the  differential 
equal  to  zero,  and  deducing  the  value  of  r.  But  according  to 
the  experiments  of  Coulomb,  the  friction  is  in  general  equal  to  -j-'jth 
of  the  resisting  pressure.  Hence  we  may  omit  /  8,  and  con- 
sider  the    resistance    as  =:  ?•  -}-  jj    r  :=   y—  r.        Consequently, 

16  »D  ,  ^  P  ^\    ..     ^'^        ry         c  J         1 

r?  r  =  4r-rj  and  r  =  (  -^^-r-  )  X  tt:-  l^ut  11  we  consider  the 
15  2  5  \  2  0  /         lb 

15 

fraction    v^  as  so  near  1,  that  the  substitution  of  the  latter  will  not 
16 

greatly  affect  the   result,  we   shall  obtain,  by  making  p  =  I  and 

D  =  1,  r  =  ^-T  ;  that  is,  the  resistance  should  be  nearly  one   half 

of  the  force  which  would  keep  the  impelling  power  in  equilibrium, 

a   rule    which    is    applicable    to    many    cases    where  the    matter 

moved  by  the  working  point  of  the  machine  is  inconsider- 
able. 


-   Maximum  Effect  of  Machines,  285 

398.  In  those  cases  where  n  varies  along  with  r,  it  will  in 
general  vary  in  the  same  proportion,  and  we  may  therefore  re- 
present n  by  X  r,  some  multiple  of  r.  For  the  sake  of  simplicity, 
the  friction  /  may  be  considered  as  absorbing  a  certain  portion 
of  the  impelling  power,  which  will  then  be  represented  by  j? — /; 
and  we  may  also  regard  the  inertia  of  the  machine,  or  i,  as  ap- 
plied at  the  impelled  instead  of  the  working  point;  that  is,  the 
moment  of  inertia  may  be  considered  as  proportional  to  i  d^. 
Now,  if  we  make  p — f  =  1,  and  d  =  1,  in  the  formula 

r  p  T>  d  —  7*2  ^2  —  r  f  6^ 
w  d2  +  n  52  -j-  i  (52      ' 

we  shall  obtain 

r  g  —  r2  32 

■and  making  m  -|-  i  =  5,  we  have 

r  8  —r'^  8^ 
s  -j-  X  r  52 

for  the  work  performed. 

This  is  a  maximum  when  the  differential,  r  being  considered  as 
variable,  is  equal  to  zero,  which  gives 

(5  _  2  r  52)    (s  +  0^  r  52)  _  a:  52  (r  5  —  r^  S^)  =  0 -, 

or,  by  reducing, 

s  5  —  2sr  8^  —  xr^  S'^  —Q; 
that  is, 

^    X  (52   —  a;  (53' 

and  by  resolving  this  after  the  manner  of  an  equation  of  the  second 
degree,  we  obtain 


-  '  +  '- 

:=  1,  we  have 

13  + 

S2 

x'^8^ 

Vj 

5^  +  . 

;2  — s 

(52 

V»  X  8  +  s^  ■>—  s 
1T2 


286 


Dynamics. 


This  case  takes  place  when  the  machine  is  employed  in  raising  a 
weight,  drawing  water,  &.c. 

When  /  =  0,f:=  0,  and  m  =  p,  then  m-\-iovs=.p:=l,  and 

_  V/(^  +  1  — 1 

When  D  =  ^,  as  in  the  common  pulley,  then  5  ~  1,  and 

r  -  ^  ^  +^  ^  ~  ^  =  V2  —  1  =  0.4142. 

In  order  to  save  the  trouble  of  calculation,  we   have  added   the 
following  table,  computed  from  the  formula 

_  y/r+l  —  1 
^—  52 

Table  containing  the  best  Proportions  between  the  Power  and  Resist- 
ance, the  Inertia  of  the  impelling  Power  being  the  same  with  its 
Pressure,  and  the  Friction  and  Inertia  of  the  Machine  being 
omitted. 


Values  of| 

'  ,     .      j  Values  of  r, 

c.\.       orthe  resist- 
or the    I  ,      , 
, .       ance   to    be 
workins; , 

■  .     °  overcome,  p 
point,    dI,    .  ^^ 

being  "^ 

equal  to  1. 


1.8885 
1.3928 
0.8986 
0.4142 
0.1830 
0.1111 
0.0772 
0.0580 
0.0457 


Ratio  of  r  to  the 
resistance  which 
would  balance  j7. 


0.4724 
0.4639 
0.4493 
0.4142 
0.3660 
0.3333 
0.3088 
0.2900 
0.2742 


to   1 


Values  ofl 
S,  or  the' 
velocity 
of  the 
working 
point,    D 

being 
equal  to  1. 


Values  of  r, 
or  the  resist- 
ance to  be 
overcome,  p 
being  =  1. 


9 
10 
11 
12 
13 
14 
15 


0.03731 
0.03125 
0.02669 
0.02317 
0.02037 
0.01809 
0.01622 
0.01466 
0.01333 


Ratio  of  r  to  the 
resistance  which 
would  balance  p. 


0.26117  to  1 

0.25000 

0.24021  

0.23170 

0.22407 

0.21708 

0.21086 

0.20524 

0.19995 


In  order  to  understand  the  method  of  using  this  table,  let  us 
suppose  tliat  it  is  required  to  find  the  value  of  the  resistance,  or 
the  quantity  of  water  which  must  be  put  into  a  bucket  to  be  raised 
by  a  wheel  and  axle,  in  which  the  radius  of  the  wheel  is  6  feet, 
and  that  of  the  axle  2  feet,  and  with  a  power  =  8.  Since,  in  the 
table,  D  =  1,  we  have 


Maximum  Effect  of  Machines. 


287 


6  :  2  :  :  D  =  1   :  5  =  g  =  3, 

which  corresponds  in  the  table  to  1,3928,  the  value  of  r  when 
p  =  1.     But,  in  the  present  case,  p  =.  8,  consequently 

1:8::   1.3928  :   11.1424, 
the  value  of  r  whea  p  =z  S. 

399.  The  subject  of  the  maximum  effect  of  machines  may- 
be considered  in  a  very  simple  point  of  view,  if  we  suppose,  what 
is  by  no  means  improbable,  that  the  moving  power  in  machinery 
observes  the  same  law  that  has  been  found  to  exist  with  regard 
to  animal   force,    and   also  with  regard  to  the  force  of  fluids  in 

(v'V 
1 )  ' 

and  the  effect  of  the  machine  will  be  r  v'  =pv'll j  , 

which  will  be  a  maximum  when  —  =z  ^  p,  and  when  v'  =z  ^  v. 

In  these  formulas,  p  is  the  load  that  is  just  sufficient  to  bring  the 
machine  to  rest,  or  prevent  it  from  moving,  v  is  the  greatest  velocity 
of  the  power  when  no  work  is  done,  and  v'  the  velocity  of  the  im- 
pelled point  of  the  machine.  The  above  equation,  both  members 
being  multiplied  by  9,  is  equivalent  to  the  proportion,  d  is  to  5,  or 
the  velocity  o£  the  impelled  point  is  to  the  velocity  of  the  working, 
point,  when  the  effect  is  a  maximum,  as  9  r  to  4  p. 


394. 


Table  of  the  Strength  of  Men,  according  to  different  Authors. 


Number  of 

Height  to  wliich 

Time  in   which  it 

Duration  of  the 

Names  of  the  Au- 

pounds    rais- 

ihe   weight    is 

is  raised. 

Work. 

thors. 

ed. 

raised. 

1000 

180  feet 

60  minutes 

Euler 

60) :? 

li   :? 

1  second 

8  hours 

Bernoulli 

2.5}  g 

220  }  g 

14.5  seconds 

Amontons 

170  j  g. 

1)  s- 

1  second 

half  an  hour 

Coulomb 

1000 

330 

60  minutes 

Desaguliers 

1000 

225 

60  minutes 

Smeaton 

30 

H 

1  second 

10  hours 

Emerson 

30 

2,43  feet 

1  second 

Schulze 

288  Dynamics. 

The  following  are  the  estimates  that  have  been  made  of  the  rel- 
ative strength  of  horses,  asses,  and  men. 

,  ,  .  1  .     c  ^  Desaejuliers. 

1  horse  is  equal  to  5  men,  <  ^       . 

^  '    (  bmeaion. 

1         "  "7  men,      Bossut,  &,c. 

1  ass  "       2  men,      Bossut. 


HYDROSTATICS. 


INTRODUCTORY  REMARKS. 

400.  Hydrostatics  is  that  part  of  Mechanics  which  treats  of 
the  equilibrium  of  fluids,  and  that  of  solids  immersed  in  them. 

A  fluid  is  a  collection  of  material  particles,  so  constituted  as 
to  yield  to  the  smallest  force  employed  to  separate  them.  The 
fluids  with  which  nature  presents  us,  approach  more  or  less  to 
this  state  of  perfect  fluidity.  The  adhesion  which  exists  among 
the  particles  of  several  of  these  substances,  and  which  gives 
rise  to  what  is  called  viscidity,  opposes  itself  to  the  separation 
of  the  particles  ;  but  in  the  theory  which  we  are  about  to  un- 
fold, no  account  is  taken  of  this  adhesion,  and  we  have  reference 
only  to  the  perfect  fluids. 

We  distinguish  two  kinds  of  fluids ;  the  one  incompressible; 
or  nearly  so,*  as  water,  mercury,  alcohol,  and  liquids  gener- 
ally. These  are  capable  of  taking  an  infinite  variety  of  forms 
without  any  sensible  change  of  bulk.  The  second  kind  of 
fluids  comprehends  atmospheric  air,  the  gases  generally,  and  va- 
pors. These  are  in  an  eminent  degree  compressible ;  they  are 
also  endued  with  a  perfect  elasticity,  and  are  thus  capable  of 
changing  at  the  same  time  their  form  and  bulk,  upon  being  com- 
pressed, and  of  recovering  their  figure  again  when  the  compress- 
ing force  is  removed.  Vapors  difier  from  air  and  the  gases,  by 
losing  the  form  of  elastic  fluids  and  returning  to  the  state  of 
liquids,    when    compressed    to    a    certain  degree,   or   when   their 

*  See  note  subjoined  to  this  treatise  on  the  compressibility  of 
water. 
Mech.  37 


290  Hydrostatics. 

temperature  is  sufficiently  reduced ;  whereas  air  and  the  gases 
are  found,  with  few  exceptions,  to  preserve  always  their  elastic 
form  in  the  state  of  the  greatest  compression  and  lowest  tempera- 
ture to  which  lliey  have  hitherto  heen  reduced.* 

401.  Although  we  are  unahle  to  assign  the  magnitude  of  the 
elementary  parts  of  fluid  bodies,  we  cannot  doubt  tliat  these  parts 
are  material,  and  that  the  general  laws  of  equilibrium  and  mo- 
tion, already  established,  are  applicable  to  them  as  well  as  to  solids. 
But  as  this  law  of  equilibrium  is  not  the  only  one  required,  some 
other  is  to  be  sought  on  which  the  equilibrium  depends. 

402.  As  an  equilibrium  consists  in  destroying  all  the  forces 
employed,  and  as  we  do  not  know  how  the  parts  of  a  fluid  trans- 
mit their  forces  among  themselves,  it  is  only  by  having  recourse 
to  experiment,  that  we  are  able  to  establish  our  first  principles. 
We  begin,  therefore,  by  stating  what  is  most  clearly  and  certainly 
known  upon  this  subject. 


Pressure  of  Fluids. 

Fig.189,       403.  Let  JIB  CD  he  a   canal  or    tube,    composed    of   three 
branches  AB,   BC,    CD,  of  equal  diameters.     Let  us  suppose 
that  a   heavy  fluid  is   poured   into   the   branch   AB ;  it  will   pass 
through  the  branch  BCinto  the  branch  CD',  and  when  we  cease 
O  pouring,    the   surface  of  the   fluid  in    the  two   branches  will   be   in 

the  same  horizontal  line,  whatever  be  the  inclination  of  the 
branch  BC.  This  is  a  fact  abundantly  established  and  univer- 
sally admitted.  We  proceed  to  make  known  the  consequences  to 
be  deduced  from  it. 

404.  If  through  any  point.  E,  taken  at  pleasure,  we  imagine 
/.-i^— _     a  horizontal  line  EF  to  pass,  it  is  evident  that  the  weight  of  the 
fluid  EBCF  contributes  nothing  to  the   support  of  the   columns 
;^/V>V"'"v'^^      AE,   DF;  and   that  consequently  the   equilibrium  would   still   be 
^      preserved,  if  the    fluid  contained  in    EBCF  were    suddenly    de- 
prived   of  its    gravity.     This   fluid,    therefore,   is   to  be  regarded 

*  See  note  on  the  condensation  of  gases  into  liquids. 


Pressure  of  Fluids.  291 

simply  as  a  medium  of  communication  between  the  columns  AE 

and  DF;  so  that  EBCF  transmits  to  the  column  DF  all  the 

pressure    it   receives   from  ^lE ;  and   reciprocally  it  transmits  to 

AE  all  it  receives  from  DF.     It  is  also  evident  that  we  should 

arrive  at  the  same  conclusions,   if  instead  of  the  columns  AE,  DF, 

we  substitute  two  forces  of  the  same  value ;  hence,  as  the  result 

is  not  affected  by  any  inclination  of  the  branch  BC,  we  conclude 

that,  if  a  fluid,  destitute  of  gravity,  be  contained  in  any  vessel,  and  F\g.l98. 

if,  having  made  an  opening  in  the  vessel,  we  apply  to  this  opening 

any  given  pressure,  the  force  thus  exerted  will  diffuse  itself  equally 

in  all  directions.  •...  »  -  ^_*-»* 

405.  Now  it  will  be  readily  seen,  not  only  that  the  pressure 
transmits  itself  equally  in  all  directions,  but  also  that  it  acts  at 
each  point  perpendicularly  to  the  surface  of  the  vessel  contain- 
ing the  fluid  ;  for  if,  acting  on  the  surface,  it  did  not  act  perpen- 
dicularly, its  effect  could  not  be  entirely  destroyed  by  the  resist- 
ance of  this  surface ;  there  would  result  therefore  an  action  up-,A| 
on  the  parts  of  the  fluid  itself,  which,  as  it  could  not  but  trans- 
mit itself  in  all  directions,  would  necessarily  occasion  a  motion 
in  the  fluid  ;  it  would  be  impossible,  therefore,  on  such  a  suppo- 
sition, for  a  fluid  to  remain  at  rest  in  a  vessel,  which  is  contrary 
to  experience. 

406.  We  hence  conclude,  that  if  the  parts  of  a  fluid  contain- Fig.  191. 
ed  in  any  vessel  ABCD,  open  toward   AD,  are  urged  by   any 
forces   whatever,   and   are  notwithstanding  preserved  in   a  state  of 
equilibrium,    these    forces    must    be   perpendicular  to    the    surface 

AD ;  for  if  there  be  an  equilibrium,  this  equilibrium  would 
still  obtain,  if  a  covering  were  applied  of  the  same  figure  with 
the  surface  AD ;  but  we  have  just  seen,  that  in  this  case  the 
forces  acting  at  the  surface  AD  must  be  perpendicular  to  this 
surface. 

407.  Accordingly,  let  us  suppose  that  the  forces  acting  on 
the  parts  of  the  fluid  are  gravity  itself;  we  shall  infer  that  the 
direction  of  gravity  is  necessarily  perpendicular  to  the  surface 
of  tranquil  fluids ;  and  that  consequently  the  parts  of  the  same 
heavy  fluid  must  be  on  a  level,  in  order  to  be  in  equilibrium,  what- 
ever be  the  figure  of  the  vessel. 


292  Hydrostatics. 

Fig.192.  408.  Let  us  now  suppose  that  the  vessel  ABCD,  being  closed 
on  all  sides,  is  filled  with  a  fluid  destitute  of  gravity,  and  that, 
having  a  very  small  opening  at  E,  we  apply  to  it  any  force ;  it 
is  evident  that  tlio  pressure  that  would  hence  be  exerted  upon 
the  plane  surface  represented  by  BC,  would  not  depend  in  any 
degree  upon  the  quantity  of  fluid  contained  in  the  vessel,  nor 
upon  the  figure  of  the  vessel ;  but  that,  since  the  pressure  applied 
404.  at  E  transmits  itself  equally  in  all  directions,  the  pressure  upon 
BC  would  be  equal  to  that  exerted  upon  any  point  of  the  opening 
E,  repeated  as  many  limes  as  there  are  points  in  BC. 

• 
409.  For  the  same  reason,  the  pressure  applied  at  E,  trans- 
mitting itself  in  all  directions,  would  tend  to  raise  the  superior 
surface  AD,  and  the  force  thus  exerted  would  be  for  each  point 
equal  to  the  pressure  applied  at  any  point  of  the  opening  E ;  so 
that  the  surface  AD  is  pressed  perpendicularly  from  within  out- 
ward with  a  force  equal  to  the  pressure  employed  at  any  point 
of  the  opening  E,  repeated  as  many  times  as  there  are  points 
in  AD. 

Fig.  193.  410.  Let  the  vessel  ABCDEF,  the  part  CD  being  horizontal, 
be  filled  with  a  heavy  fluid.  We  say  that  the  pressure  upon  the 
bottom  CD,  arising  from  the  gravity  of  the  fluid,  does  not  depend 
upon  the  quantity  of  fluid  contained  in  the  vessel,  but  simply 
upon  the  extent  of  CD,  and  its  depth  below  the  surface  AF. 

To  make  this  evident,  let  us  suppose,  the  line  BE  being  hori- 
zontal, that  the  fluid  contained  in  BCDE,  is  suddenly  deprived 
of  its  gravity,  it  is  evident  that  a  vertical  filament  IK,  of  heavy 
particles  of  the  fluid  contained  in  ABEF,  would  exert  at  the 
point  K  a  pressure  which  must  diffuse  itself  equally  throughout 
the  whole  extent  of  the  fluid  B  CDE  ;  that  this  pressure  would  be 
^^^-  exerted  with  equal  force  from  below  upward  to  repel  the  action 
of  each  of  the  other  vertical  filaments  belonging  to  the  several 
points  of  BE;  hence  the  filament  JBT effects,  by  itself,  an  equili- 
brium with  all  the  other  filaments  of  the  mass  ABEF;  therefore 
the  mass  BCDE  being  destitute  of  gravity,  there  will  result  no 
other  pressure  on  the  bottom  CD,  than  that  arising  from  the  fil- 
ament IK,  which  transmitting  itself  equally  to  all  the  points  of 
CD  causes   upon    CD  a  pressure   equal  to  that  exerted  at  the 


Pressure  of  Fluids.  293 

point  K,  repeated  as  many  times  as  there  are  points  in    CD.     Ac-  Fig.194. 

cordingly,  if  we  suppose  the  heavy  fluid  contained  in  ACDF, 
divided  into  horizontal  strata,  the  upper  stratum  would  commu- 
nicate to  the  bottom  CD  no  other  action  than  that  which  would 
be  communicated  by  the  filament  a  h  ;  and  the  same  being  true 
of  each  stratum,  the  bottom  CD  will  only  receive  the  pressure 
that  would  arise  from  the  sum  of  the  filaments  a  b,  b  c,  c  d,  he.  ; 
and  since  this  pressure  would  transmit  itself  equally  to  all  the 
points  of  CD,  it  is  equal  to  the  area  of  CD  multiplied  by  the 
sum  of  the  pressures  exerted  upon  some  one  point  by  the  sum 
of  the  filaments  a  b,  b  c,  c  d,  he. ;  from  all  which  we  derive  the 
following  conclusions,  namely ; 

(1.)  If  the  fluid  ACDF  be  homogeneovs,  that  is,  composed  of  parts 
of  the  same  nature,  of  the  same  gravity,  <^c.,  the  pressure  upon  the 
bottom  CD  ivill  be  expressed  by  CD  X  a  g ;  or,  in  other  words, 
ivill  be  measured  by  the  weight  of  the  prism  or  cylinder  ivhich  has 
CD  for  its  base  and  a  g  for  its  altitude. 

(2.)  if  the  fluid  is  composed  of  strata  of  different  densities,  the 
pressure  upon  CD  tmll  be  expressed  by  CD  multiplied  by  the  sum  of 
the  specific  gravities  of  each  stratum ;  I  say  by  the  sura  of  the  spe- 
cific gravities,  and  not  by  the  sum  of  the  weights  ;  for  it  is  not 
on  the  quantity  of  the  fluid  contained  in  each  stratum  that  the 
pressure  depends  but  simply  on  the  proper  gravity  of  each  fil- 
ament. 

It  is  important  to  observe  that  the  above  propositions  hold 
true,  whether  the  vessel  grows  larger  toward  the  top,  as  in  the 
present  instance,  or  whether  it  is  constructed  from  the  bottom 
upw^ard,  as  represented  in  figure  195.  The  |)ressure  which  the 
fluid  contained  in  .4  CD i^  exerts  upon  CD,  is  the  same  as  if  the 
cylinder  ECDG  were  filled  with  the  fluid,  the  altitude  being  the 
same  in  both  cases.  This  constitutes  what  is  called  the  hydro- 
static paradox,  and  is  often  expressed  in  the  following  words, 
namely;  any  quantity  of  water  or  other  fluid,  however  sinall,  may 
be  made  to  balance  and  support  a  quantity  however  large.  The 
principle  is  well  illustrated  by  an  instrument  called  the  hydro- 
static bellotvs ;  see  figure  196,  in  which  jEi^,  CD  represents  two 
thick  boards  IG  or  18  inches  in  diameter,  firmly  connected  to- 
gether by  pliable  leather  attached  to  the    edges,  which  allows  a 


294  >  Hydrostatics. 

motion  like  that  in  common  bellows.  Instead  of  a  valve,  a  pipe 
jIB,  about  three  feet  in  length,  is  inserted  at  B,  either  in  the 
upper  or  lower  part  of  the  bellows.  Now  if  water  be  poured 
into  this  pi|)e  at  ^1,  it  will  descend  into  the  bellows  and  grad- 
ually separate  the  pieces  EF,  CD,  from  each  other  by  rais- 
ing the  latter ;  and  if  several  weights,  200  pounds,  for  instance, 
be  placed  upon  the  upper  board,  the  small  quantity  of  water  in 
the  pipe  ^B  will  balance  all  this  weight.  More  water  being 
poured  in,  instead  of  filling  the  pipe  and  running  over  the  lop  ^, 
it  will  descend  into  the  bellows,  and  slowly  raise  the  weights ; 
the  distance  between  the  surface  of  the  water  in  the  pipe  and 
that  in  the  bellows  remaining  the  same  as  before.  It  is  manifest 
from  what  has  been  said,  that  the  upward  pressure  exerted  upon 
each  point  of  the  interior  surface  o(  CD  is  sufficient  to  support  a 
column  of  fluid  of  the  same  height  with  that  contained  in  the  tube 
AB,  and  consequently  that  the  whole  upward  pressure,  or  weight 
sustained,  is  equal  to  the  weight  of  a  cylinder  of  water,  whose 
base  is  the  area  of  CD,  and  whose  altitude  is  that  of  the  column 
of  water  in  AB  above  the  surface  of  the  water  in  the  bellows  or 
lower  surface  of  CD.  The  area  of  the  base,  for  instance,  being 
a  foot  and  a  half,  and  the  altitude  three  feet,  the  w^hole  mass 
would  be  4|-  cubic  feet,  and  the  weight  sustained  would  be 
4^  X  62|^*  or  28  li  pounds,  while  the  quantity  contained  in  AB, 
depending  on  the  size  of  the  tube  might  weigh  only  one  fourth  of 
a  pound  or  any  less  quantity. 

It  is  obvious  that  instead  of  the  gravity  of  the  fluid  in  the 
tube  AB,  any  other  force  might  be  employed,  as  the  impulse  of 
the  breath,  or  that  exerted  by  a  stopper  or  piston  moving  in  the 
tube  AB.  Thus,  by  means  of  a  lever  HI,  a  dense  fluid,  as  water 
Fig.197.  for  instance,  might  be  forced  through  the  pipe  CO  against  a 
large  piston  supposed  to  be  accurately  fitted  to  the  cylinder  FD, 
and  connected  with  the  rod  or  bar  DE.  A  valvef  being  provid- 
ed at  jP  to  prevent  the  return  of  the  fluid,  this  action  might  be 
repeated ;  we   should   thus   have    an    engine   of  almost   unlimited 


*A  cubic   foot  of  water   at  the   temperature  of  50°  weighs 
1000°*-  avoirdupois  or  62i^>'-. 

t  See  note  on  the  construction  of  valves. 


Pressure  of  Fluids.  295 

power  contained  at  the  same  time  within  a  small  compass,  and 
very  simple  in  its  construction.  The  diameter  of  the  large  pis- 
ton being  12  inches,  for  example,  and  that  of  the  small  one  worked 
by  the  lever  H  and  moving  in  jIB,  only  one  fourth  of  an  inch, 
the  proportion  of  the  two  surfaces,  or  of  the  power  employed  to 
the  force  exerted  at  E,  would  be  as  Jg  to  144,  or  as  1  to  2304. 
Now  it  would  be  easy,  by  means  of  the  lever  HI,  to  apply  to  the 
small  piston  a  force  equal  to  20  cwt,  or  one  ton,  in  which  case 
the  piston  working  in  FD  would  be  moved  with  a  force  of  2304 
tons.  This  instrument  is  called  the  Hydrostatic  or  BraniaK's 
Press,  Mr.  Bramah,  an  Englishman,  being  the  first  person  who 
made  use  of  the  hydrostatic  principle  here  involved,  as  a  substi- 
tute for  (he  screw  in  the  construction  of  presses.*  This  machine 
evidently  belongs  to  the  class  of  mechanical  powers,  and  is  es- 
sentially different  in  its  nature  from  those  heretofore  described. 
The  principle  of  virtual  velocities,  however,  is  equally  applicable 
to  this ;  since  the  greater  the  advantage  gained  in  point  of 
intensity,  just  so  much  is  lost  in  respect  to  velocity.  Suppose, 
for  example,  that  the  pipe  AB,  filled  with  fluid,  is  2304  inches  in 
length,  the  small  piston,  by  moving  through  this  whole  extent, 
and  thus  forcing  the  entire  contents  of  the  pipe  into  the  cylinder 
FD,  would  raise  the  large  piston  only  one  inch  ;  so  that  while 
the  pressure  upon  the  small  piston  is  to  that  upon  the  large  one, 
in  case  of  an  equilibrium,  as  1  to  2304,  the  spaces  described  in 
the  same  time,  or  the  velocities  of  the  two  pistons,  are  as  2304 
to  1,  and  the  quantity  of  motion  in  each  is  the  same. 

411.  Let  there  be  two  fluids  JVHC^FL,  EFL^k?  of  difi'erentFig.igs. 
densities,  but  each  being  homogeneous,  considered  by  itself,  and 
let  them  be  made  to  act  against  each  other  at  FL  by  means  of 
the  vessel  in  wliich  they  are  contained.  They  can  be  in  equi- 
librium only  when  the  altitudes  EF,  IK,  above  the  horizontal 
plane  FL  which  separates  them,  are  inversely  as  their  specific 
gravities.  Indeed,  the  fluid  LFBCGO  being  itself  in  equilibrium,  407. 
it  is  necessary  that  JVHGO  should  be  in  equilibrium  with  EFLM; 
it   follows,    therefore,    that  the   upward    pressure  exerted    by  the 

*  The  same  property  of  fluids  is  sometimes  employed  very  ad- 
vantageously in  a  crane  and  in  raising  water  froru  mines. 


Fig.191 


296  Hydrostatics. 

column  JVHGO  upon  FL,  should  be  equal  to  the  downward 
pressure  exerted  upon  FL  by  the  column  EFLM.  Now  the 
pressure  of  JVHGO  upon  FL  is  equal  to  the  weight  of  a  prism  or 
cylinder  of  this  fluid  which  has  the  surface  FL  for  its  base  and  IK 
for  its  altitude  ;  moreover  this  weight  is  equal  to  the  specific 
gravity  multiplied  by  the  bulk ;  accordingly,  if  we  call  the  spe- 
cific gravity  S,  we  shall  have  for  the  expression  of  the  weight 
S  X  IK  X  FL.  For  the  same  reason,  if  we  call  S',  the  specific 
gravity  of  the  fluid  EFLM,  we  shall  have 

.5''  X  jEF  X  FL, 

as  the  expression  for  the  absolute  gravity  of  this  fluid  or  the  pressure 
which  it  exerts  upon  FL.     Therefore 

S  X  IKx  FL  =  S'  X  EFx  FL, 

or 

S  X  IK=S'  X  EF; 

whence 

S  :  S'  ::  EF  :  IK, 

that  is,  the  altitudes  are  inversely  as  the  specific  gravities.  Thus 
if  LFBCHN  were  mercury,  and  EFLM  water  ;  since  mercury 
is  13,6  or  nearly  14  times  as  heavy  as  water,  the  altitude  IK 
would  be  one  fourteenth  part  of  EF,  whatever  be  the  figure  of  the 
vessel. 

412.  From  what  has  been  said,  it  will  be  seen  that  the  action 
of  fluids  is  very  difierent  from  that  of  solids.  Properly  speaking, 
it  is  only  the  part  ECDG  which  exerts  its  action  upon  the  surface 
CD;  and  in  figure  195,  the  surface  CD  is  pressed  by  A  CDF,  as 
it  would  be  by  the  weight  of  the  fluid  contained  in  the  cylinder 
ECDG.  If,  on  the  contrary,  the  fluid  ACDF  were  suddenly  to 
become  a  solid,  by  freezing,  the  bottom  would  support  a  pressure 
equal  to  the  weight  of  the  entire  mass  ACDF  in  figure  194,  and 
only  equal  to  the  weight  o[  ACDF in  figure  195. 

413.  It  is  necessary  here  to  distinguish  between  the  force  or 
pressure  exerted  on  the  bottom  CD,  and  thai  which  would  be 
sustained  by  a  person  carrying  the  vessel.  It  is  clear  that  if  the 
bottom  CD  were  movable,  the  only  thing  necessary  to  keep  it 
in  its  place,  would  be  an  eflbrt  equal  to  the  weight  of  the  cylin- 


Pressure  of  Fluids,  297 

der  ECDG,  but  in  order  to  transport  the  vessel,  an  effort  would  Fig.i94. 
be  required  equal  to  the  weight  of  the  entire  mass  of  water  con- 
tained in  the  vessel.  This  will  be  demonstrated  in  a  manner  still 
more  general  after  we  have  explained  the  method  of  estimating 
the  pressure  upon  oblique  plane  surfaces  and  upon  curved  sur- 
faces. 

414.  Let  ACDFhe  the  vertical  section  of  a  vessel  terminated  Fig.  199, 
by  surfaces   either  plane   or   curved,  and  inclined  in   any  manner 

to  the  horizon.  If  we  imagine  an  infinitely  thin  stratum  a  b  d  c, 
we  can  suppose  it  destitute  of  gravity,  and  pressed  by  the  fluid 
above  it.  Now  this  pressure  will  be  distributed  equally  to  all 
points  of  the  stratum,  and  will  act  perpendicularly  and  equally 
upon  each  of  the  points  of  the  faces  a  c,  b  d.  Accordingly,  as 
this  force  is  equal  to  that  which  a  single  filament  IK  would 
cause,  the  pressure  exerted  perpendicularly  upon  b  d,  will  be 
expressed  by  b  d  X  IK;  and  it  is  evident  that  we  should  arrive 
at  the  same  result,  if  instead  of  6  c?  being  considered  as  a  small 
straight  line,  we  regard  it  as  a  small  surface.  We  hence  derive 
the  general  conclusion,  that  the  pressure  exerted  perpendicularly 
upon  any  infinitely  small  surface  by  a  heavy  homogeneous  fluid,  has 

for  its  expression  this  surface  multiplied  by  its  perpendicular  distance 

from  the  level  of  the  fluid. 

415.  Hence  the  whole  pressure  exerted  upon  any  plane  sur- 
face, situated  as  we  please,  is  equal  to  the  sum  of  the  infinitely 
small  parts  of  this  surface,  multiplied  each  by  its  distance  from 
the  level  of  the  fluid.  If  we  represent  these  small  parts  by 
m,  n,  0,  he,  and  their  distances  respectively  from  the  level  of 
the  fluid  by  AA',  BB',  CO,  &;c.,  according  to  article  76,  we  shall 
have 

GG'  X  (w  +  /I  -h  0  4-  &:c.,) 

=  AA'  X  m  -J-  BB'  X  n  +  CC  X  0  +  &:c., 

that  is,  the  sum  of  these  products  is  equal  to  the  whole  surface 
multiplied  by  the  distance  of  its  centre  of  gravity  from  the  same 
horizontal  plane.  Therefore  the  pressure  exerted  by  a  heavy  fluid 
against  an  oblique  plane  surface  has  for  its  measure  the  product  of 
this  surface  into  the  distance  of  its  centre  of  gravity  from  the  line 
of  level  of  the  fluid. 
Mech.  38 


298  Hydrostatics. 

416.  As  the  pressures  exerted  upon  the  several  points  of  the 
same  plane  surface  are  perpendicular  to  the  surface,  and  conse- 
quently parallel  among  themselves,  the  resultant  or  whole  press- 
ure must  be  parallel  to  the  components.  Now  as  we  know  how 
to  find  the  resultant  as  well  as  that  of  each  of  the  partial  press- 
ures, it  will  be  easy  to  determine,  as  we  have  occasion,  through 
what  point  the  resultant  passes ;  it  evidently  cannot  pass  through 
the  centre   of   gravity,  hut  must  pass  through   some    point  lower 

Fig. 201.  down.  It  is  only  in  the  case  where  the  surface  is  infinitely  small 
that  the  whole  pressure  passes  through  the  centre  of  gravity  of 
this  inclined  surface. 

417.  In  order  to  find  the  resultant  of  all  these  pressures  both 
in  a  vertical  and  in  a  horizontal  direction,  any  body,  whatever 
its  figure,  may  be  considered  as  composed  of  an  infinite  number 
of  strata,  parallel  among  themselves,  and  the  surface  of  the  pe- 
rimeter of  each  stratum  may  be  represented  by  a  series  of  tra- 
pezoids, of  which  the  number  is  infinite,  when  the  surface  is  a 
curve.  So  that  in  order  to  estimate  the  resultant  of  the  pressure 
exerted  by  a  fluid  either  upon  the  interior  sides  of  a  vessel,  or 
upon  the  exterior  surface  of  a  solid  immersed  in  it,  we  must  de- 
termine the  pressure  exerted  upon  a  trapezoid  of  an  infinitely 
small  altitude. 

Accordingly,  let  us  suppose  a  trapezoid  ABCD,  of  which  the 
Fig.202.  two  parallel  sides  are  AB,  CD,  and  the  altitude  infinitely  small 
compared  with  these  sides ;  and  let  there  be  applied  at  the  cen- 
tre of  gravity  G  of  the  trapezoid  perpendicularly  to  its  plane,  a 
force  p  equivalent  to  the  product  of  the  surface  of  this  trapezoid 
into  the  distance  of  its  centre  of  gravity  from  the  horizontal 
plane  XZ. 

To  determine  the  effect  of  this  force,  as  well  in  a  horizontal 
as  in  a  vertical  plane,  suppose  through  the  line  CD  a  vertical 
plane  CDFE,  and  through  the  line  AB,  considered  as  horizontal, 
a  horizontal  plane  AFEB.  Having  drawn  the  vertical  lines 
CE,  DF,  meeting  this  latter  plane  in  E  and  F,  we  join  BE,  AF ; 
and  through  the  direction  G  p  of  the  force  p,  suppose  a  plane 
KIH  cutting  CD  at  right  angles,  HGK  and  HI  being  the  inter- 
sections of  this  plane  with  the  two  planes  ABCD,  FECD,  re- 
spectively.    The.  plane  KIH   will    be  perpendicular   to  each  of 


Pressure  of  Fluids.  299 

the  two  planes  ABCD,  FECD,  since  CD  is  their  common  inter- Geom. 
section ;  lastly,  from  the  point  K,  where  HK  meets  AB,  let  fall 
the  perpendicular  KI  upon  the   plane   lECD,  this  line  must  be 
perpendicular  to  HI. 

These  steps  being  taken,  I  decompose  the  force  p  into  two 
others,  both  being  in  the  plane  KIH  produced,  and  of  which  one 
GL  is  horizontal  or  perpendicular  to  the  plane  FECD,  and  the 
other  GM  vertical.  Calling  these  two  forces  q  and  r,  and  forming 
the  parallelogram  GMNL  upon  the  line  GN,  taken  arbitrarily  as 
the  diagonal,  we  shall  have 

p  :  q  :  r  :  :  GN  :   GL  :  GM, 
::  GN  :  GL  :  LN. 

But  as  the  triangle  GLN  has  its  sides  perpendicular  respectively 
to  those  of  the  triangle  KIH^  these  two  triangles  are  similar,  and 

we  have  Geom. 

209. 

GN  :  GL  :  LN  :  :  HK  :  HI  :  IK; 

and  consequently 

p  :  q  :  r  ::  HK  :  HI  :  IK. 

Multiplying  the  three   last  terms  by  ~ X    GG',  which 

does  not  change  the  ratio,  we  obtain, 

p  :  q  :  r 

::  HKx  ^^  ^  ^^  X  GG'  :  HI 
X  d£_±£5  X  GG'  :  IK  X    ^-^  X  GG'. 

We  observe  now,  (1.)  That  HST  X  ~ is  the   surface   of 

the  trapezoid  ABCD;   (2.)  That  since  CE,  DF  are  parallel,  as 
also  CD,  EF,   CD  is  equal  to  FE;  whence  IK  X  ^^  +  ^^  is 

Geom. 
.     ,                 „^          AB  4-  EF  ,  ,       .        ,  178. 

equivalent  to  IK    X  s >    ^nt'    consequently    is   the   sur- 
face of  the  trapezoid  AFEB  ;    (3.)  As  we  suppose  the  altitude  of 


300  Hydrostatics. 

the  trapezoid  ABCD  infinitely  small  compared  with  the  sides 
AB  and    CD,  EF  which  is   equal  to  CD,  may  be  taken  instead 

of  AB  and  also  instead  of  CD,  so  that  HI  X  -^J reduces 

itself  to  HI  X  ^^^^^—  =  HI  X  ^F",  which  is  the  surface  of  the 

rectangle  ECDF;  we  have,  therefore, 

p:q:r::  ABCD  X  GG'  :  ECDF  X  GG^'  :  AFEB  X  G'G'. 

But  we  have  supposed  the  force  p  expressed  by  ABCD  X  GG', 
hence  the  force  g  will  be  expressed  by  ECDF  X  GG',  and  the 
force  7- by  ^Fi;^  X  GG\ 

Since  a  triangle  is  simply  a  trapezoid,  one  of  whose  parallel 
sides  is  zero,  the  same  results  are  applicable  to  a  triangle. 

Suppose  now  perpendiculars  let  fall  from  the  angles  A,  D,  C, 
B,  upon  the  plane  XZ.  These  perpendiculars  may  be  consider- 
ed as  the  edges  of  a  truncated  prism,  the  horizontal  base  being 
AFEB,  and  the  inclined  base  ABCD.  Now  as  AB,  CD,  are 
supposed  to  be  infinitely  near  to  each  iJther,  the  bulk  of  this 
prism  may  be  regarded  as  not  differing  from  that  of  a  prism  of 
the  same  base,  and  whose  altitude  is  GG' ;  but  this  last  has  for  its 
expression  AFEB  X  GG^,  which  is  precisely  that  just  found  for 
the  vertical  force  r ;  therefore  this  force  has  also  for  its  expression, 
the  bulk  of  the  truncated  prism,  whose  inclined  base  is  ABCD 
and  whose  horizontal  base  is  the  projection  of  ABCD  upon  the 
horizontal  plane  XZ. 

418.  Let  any  solid  be  divided  into  an  infinite  number  of  hor- 
Fig.203.izontal  strata  ABDE  a  b  d  e,  and  suppose  that  at  the  centre  of 
gravity  of  the  surface  of  each  trapezoid  of  which  the  surface  of 
the  perimeter  of  this  stratum  is  composed,  forces  are  applied, 
represented  each  by  the  surface  of  the  corresponding  trapezoid 
multiplied  by  the  distance  of  the  centre  of  gravity  from  a  hori- 
zontal plane  XZ.  These  forces  will  represent  the  pressure  ex- 
4J5^  erted  by  a  heavy  fluid  upon  the  interior  surface  of  the  stratum 
ABDE  a  b  d  e  o(  B.  vessel  in  which  this  fluid  is  contained ;  they 
will  also  represent  the  pressure  exerted  by  a  similar  fluid  upon 
the  exterior  surface  of  a  solid  immersed  in  this  fluid.  Now  we 
have    seen    that    these    forces    being  decomposed   each   into   two 


Pressure  of  Fluids.  301 

others,  one  vertical  and  the  other  horizontal,  each  vertical  force  will 
be  represented  by  the  truncated  prism  which  has  for  its  base  the 
projection  of  the  trapezoid  upon  the  horizontal  plane  XZ,  and  for 
its  inclined  base,  this  trapezoid  itself.  Therefore  the  sum  of  the 
vertical  forces,  or  the  single  vertical  force  that  would  result  from 
them,  will  be  represented  by  the  sum  of  all  the  truncated  prisms ; 
and  as  the  same  reasoning  is  applicable  to  each  horizontal  stratum,  we 
conclude,  (1.)  That  if  a  vessel  ABCDF,  of  any  figure  whatever,  be 
filled  with  a  fluid  to  any  line  AF,  there  will  result  from  all  the 
j)ressnres  exerted  by  this  fluid  upon  the  several  points  of  the  vessel, 
no  other  vertical  force  than  that  ivhich  is  represented  by  the  bulk 
of  this  fluid,  or  rather  by  its  iveight. 

(2.)  That  if  a  body,  as  ACDBM,  for  example,  of  which  AIBF  Fig.204. 
is  the  greatest  horizontal  section,  be  immersed  in  a  fluid  to  any 
depth  whatever,  the  pressure  exerted  upon  the  superior  part  AMB 
being  left  out  of  consideration,  the  vertical  effort  of  the  fluid  to  raise 
the  body,  is  equal  to  the  weight  of  a  volume  of  this  fluid  compre- 
hended between  the  level  XZ,  the  surface  AIBFC,  and  the  convex 
surface  terminated  by  the  perpendiculars  let  fall  from  the  several 
points  of  the  perimeter  AIBF  upon  the  plane  XZ. 

If  we  next  consider  the  pressure  exerted  upon  the  surface  above 
the  greatest  horizontal  section,  it  will  be  seen,  by  the  same  kind  of 
reasoning,  that  there  would  result  from  the  pressure  of  the  fluid 
upon  this  surface  in  a  vertical  direction,  a  downward  effort  equal 
to  the  weight  of  a  bulk  of  the  fluid  comprehended  between  this 
same  surface,  that  of  its  projection  A'F'B'l',  and  that  terminated 
by  the  perpendiculars  let  fall  from  the  several  points  of  the  pe- 
rimeter AIBF.  Accordingly,  if  from  the  first  vertical  effort,  we 
subtract  the  second,  it  will  be  seen  that  the  body  is  urged  ver- 
tically upward  by  an  effort  equal  to  the  weight  of  a  bulk  of  this 
fluid  of  which  it  occupies  the  place. 

419.  We  hence  derive  the  general  conclusion,  that  if  a  body 
be  immersed  in  any  fluid  whatever,  it  loses  a  part  of  its  weight  equal 
to  the  weight  of  the  fluid  displaced,  or  equal  to  the  weight  of  its  own 
bulk  of  this  fluid.  ^t^ 

420.  There  remain  now  two  things  to  be  inquired  into  ;  the 
first  is,  to  determine  through  what  point  the  vertical  effort,  result- 


302  Hydrostatics. 

ing  from  the  pressure  of  the  fluid,  passes ;  the  second,  to  find  what 
becomes  of  the  horizontal  forces. 

(1.)  It  will  be  seen  that  the  vertical  effort  must  pass  through 
the  centre  of  gravity  of  the  portion  of  fluid  displaced.  For,  if 
we  imagine  this  portion  decomposed  into  an  infinite  number  of 
vertical  filaments,  the  vertical  effort  which  the  fluid  exerts  upon 
each  filament,  is  expressed  by  the   weight  of  a  portion   of  fluid 

272.  equal  to  this  filament ;  consequently,  to  find  the  distance  of  the 
resultant  from  any  vertical  plane,  it  is  necessary  to  multiply  the 
mass  of  each  filament,  considered  as  of  the  same  nature  with  this 
fluid,  by  its  distance  from  this  plane,  and  to  divide  by  the  sum  of 
the  filaments.     But  this  is  precisely  the  course  to  be  pursued,  in 

76.  order  to  find  the  distance  of  the  centre  of  gravity  of  the  portion  of 
fluid  displaced  ;  therefore  the  vertical  effort  of  a  fluid  upon  a 
body  immersed  in  it,  passes  always  through  the  centre  of  gravity 
of  the  portion  of  fluid  displaced,  which  may  be  called  the  centre  of 
buoyancy. 

421.  (2.)  We  proceed  now  to  consider  the  horizontal  forces 
above  referred  to.  Representing  always  the  solid  stratum  by 
figure  203,  if  through  the  sides  a  b,  b  c,  &,c.,  of  the  inferior  sec- 
lion,  we  suppose  verfical  planes  to  pass  terminating  in  the  supe- 
rior section ;  these  planes  will  form  the  contour  of  a  prism  whose 
altitude  is  that  of  the  stratum  ;  and  each  face  of  this  prism  will 
417.  express  by  the  extent  of  its  surface  the  value  of  the  horizontal 
force  perpendicular  to  it.  But,  since  all  these  faces  are  of  the 
same  altitude,  their  surfaces  will  be  as  their  bases  ah,  be,  &c., 
Geo«.  consequently  the  horizontal  forces  are  to  each  other  as  the  sides 
a  b,  be,  &LC.  Moreover,  at  whatever  point  of  these  faces  they 
are  applied,  as  these  faces  are  of  an  altitude  infinitely  small,  the 
horizontal  forces  may  be  considered  as  applied  each  in  the  hor- 
izontal plane  a  b  c  d  e  f,  perpendicularly  to  the  middle  of  the  side 
which  serves  as  a  base  to  the  corresponding  face  of  the  prism  in 
question.  I  say  to  the  middle,  since  it  will  readily  be  seen  that 
the  resultant  of  the  pressures  exerted  upon  the  surface  of  any  one 
of  the  trapezoids  which  form  the  surface  of  the  stratum,  must 
pass  through  some  point  of  the  line  joining  the  middle  points  of  the 
two  parallel  sides,  and  that,  accordingly,  the  horizontal  force  ob- 
tained by  decomposing  this  resultant,  must  meet  the  line  joining 


175. 


Pressure  of  Fluids.  303 

the  middle  points  of  the  two  opposite  sides  of  the  corresponding 
face  of  the  prism.  The  problem,  therefore,  is  reduced  to  finding 
what  must  take  place  in  any  polygon,  when  each  of  its  sides  is 
drawn  or  pushed  by  a  force  applied  perpendicularly  to  its  middle 
point,  and  represented  as  to  its  value  by  this  side.  We  shall  see 
that  they  will  mutually  destroy  each  other. 

Let  usj  in  the  first  place,  consider  only  the  two  forces  p  and  g-.  Fig.  205. 
applied  perpendicularly  to  the  middle  points  of  the  sides  AB, 
A.C,  of  the  triangle  ABC,  these  forces  being  represented  as  to 
their  values  by  these  sides  respectively.  It  is  clear  that  their 
resultant  would  pass  through  their  common  point  of  meeting  F, 
which,  in  the  present  case,  is  the  centre  of  a  circle  in  whose  cir-  Geom. 
cuinference  the  points  A,  B,  C,  are  situated.  We  say,  moreover, 
that  this  resultant  will  pass  through  the  middle  point  of  BC,  to 
which  it  will  consequently  be  perpendicular,  and  that  it  will  be  rep- 
resented in  magnitude  by  BC.  For,  if  we  decompose  the  force  p 
into  two  others,  one  D  e  parallel,  and  the  other  D  h  perpendicular, 
to  BC,  by  forming  the  parallelogram  D  e  ^  A  we  shall  have,  by 
calling  these  two  forces  e  and  h  respectively, 

p  :  e  :  h  :  :  D  g  :  D  e  :  D  h  :  :  D  g  :  D  e  :  g  e. 

Now,  by  letting  fall  the  perpendicular  AO,  the  triangle  g  e  D  is 
similar  to  the  triangle  AOB,  since  dieir  sides  are  respectively  per- 
pendicular.    Accordingly, 

D  g  :  D  e  :  g  e  :  :  AB  :  AO  :  BO, 

whence 

p  :  e  :  h  ::  AB  :  AO  :  BO. 

But,  by  supposition,  the  value  of  the  force  p  is  represented  by 
AB ;  therefore  that  of  e  is  represented  by  AO,  and  that  of  h 
by  BO. 

If  we  decompose  in  like  manner  the  force  g  into  two  others, 
the  one  I  m  parallel,  and  the  other  J  A;  perpendicular,  to  jBC,  it 
may  be  shown  as  above,  that  m  is  represented  by  AO,  and  k  by 
CO.  The  two  forces  m  and  e  are  therefore  equal,  since  they 
are  represented  by  the  same  line  AO.  Moreover  they  act  in 
opposite  directions,  and  according  to  the   same  line  Dl  parallel 


304  Hydrostatics. 

to  BC,  since  D,  i,  are  the  middle  points  oi  AB.AC,  respectively. 
Consequently,  they  will  destroy  each  other.  The  resultant  then 
must  be  the  same  as  that  of  the  two  forces  h  and  k ;  and  as  these 
are  parallel,  being  each  perpendicular  to  CB,  their  resultant  must  be 
equal  to  their  sum,  and  perpendicular  lo  BC',  that  is,  (I)  It  will  be 
represented  by  BO  -\-  OC  or  by  BC;  and  (2.)  being  perpendicu- 
lar to  BC,  and  passing,  as  we  have  just  seen,  through  the  centre  F 
of  the  circle  circumscribed  about  ABC,  it  passes  through  the 
middle  point  BC. 

Fig.206.  This  being  premised,  the  resultant  q  of  the  two  forces  p,  q', 
will  be  perpendicular  to  the  middle  of  BE,  and  be  represented 
by  BE.  For  the  same  reason,  the  resultant  q'  of  the  two  forces 
Q,  f}',  and  consequently  of  the  three  forces  p,  q',  p',  will  be  per- 
pendicular to  the  middle  of  BD  and  be  represented  by  BD. 
Lastly,  the  resultant  q"  of  the  forces  q',  q,  and  consequently  of 
the  forces  p,  q',  p',  q,  will  be  perpendicular  to  the  middle  of  DC 
and  be  represented  by  DC;  it  will  accordingly  be  equal  and 
directly  opposite  to  the  force  r ;  therefore  all  these  forces  will 
destroy  each  other.  The  same  reasoning  will  evidently  be  apj)li- 
cable,  whatever  the  number  and  magnitude  of  the  sides.  Hence 
we  derive  the  general  conclusion,  that  the  efforts  which  result  in  a 
horizontal  direction  from  the  pressure  of  a  heavy  fluid  exerted  per- 
pendicidarly  upon  the  surface  of  a  body  immersed  in  it,  mutually 
destroy  each  other. 

422.  The  pressures  upon  any  given  surface  being  considered  by 

themselves,  the   distance   at  which  the  resultant,   or  the  centre  of 

pressure  passes,  is  readily  found.     The   forces  exerted   upon   the 

several   points  being  as  the  distances  respectively  of  these  points 

from  the  surface  of  the  fluid,  they  are  as  the  forces  that  would  arise 

from  the  motion  of  this  surface  about  the   intersection  of  its  plane 

with  the  surface  of  the  fluid  as  an  axis.     But  we  have  seen  that  the 

distance  of  the  resultant  in  this  case  is  that  of  the  centre  of  per- 

357.     cussion  or  oscillation.     Hence  the  distance  of  the  centre  of  pressure 

of  any  given  surface  from  the  surface  of  the  fluid,  is  the  same  as 

that  of  the  centre  of  percussion.    Accordingly,  the  centre  of  pressure 

Fig.  176.  on  the  side  of  a  perpendicular  prismatic  vessel,  is  one  third  of  the 

365.    height  from  the  bottom. 


Pressure  of  Fluids.  305 

'^^  Moreover,  since  the  magnitude  of  this  pressure  is  equal  to  the 
surface  multiplied  by  the  distance  of  the  centre  of  gravity,  the  pres-  415. 
sure  on  the  upright  side  of  a  vessel  of  the  form  of  a  parallelopiped 
is  to  the  pressure  on  the  bottom,  as  half  the  area  of  the  sideF'g'l76. 
to  the  area  of  the  bottom,  or  as  half  the  height  of  the  side 
to  the  length  of  the  bottom,  reckoned  in  the  direction  of  a  perpen- 
dicular to  this  side.  When  the  parallelopiped  is  a  cube,  the  pres- 
sure on  the  side  is  half  that  on  the  bottom,  and  the  pressure  upon 
the  four  sides  together,  double  that  upon  the  bottom. 

Since  the  pressure  of  any  fluid  is  proportional  to  the  depth 
below  the  surface,  the  strain  upon  the  sides  of  a  sluice,  and  the 
banks  of  a  canal,  must  increase  uniformly  from  the  top  to  the  bot- 
tom ;  and,  when  this  force  is  to  be  resisted  by  earth  or  masonry, 
since  the  strength  of  the  materials  may  be  estimated  in  a  certain 
proportion  to   the  weight,   commonly   as  a  third  or  fourth  part,  if  /^ 

AB  be  the  height  of  the  bank  or  dike,  and  a   third  of  its    density  Fig.  2(j|^ 
be  to  that  of  water,  as  AB  to  BC,   by  joining  AC,  this  line  will 
represent   the   proper  slope.     If  the  bank  be  composed   of   stone 
or   brick,   the  base  BC  must  be    at   least   equal    to    the    altitude |^ 
AB ;  if  it  be  of  earth,  BC  should  exceed  the   height  AB  by  one 
half. 

Let -4  C,  5  C,  represent  the  floodgates  of  a  canal-lock  which  Fig.209. 

are  opened  by  means  of  the  extended  arms  AF,  BF.     When  we 

shut  the  gate,   AC  is  pressed   at  right  angles  by   the  water  with 

a  force   proportional  to  AC,  and  which,  from  the  principle  of  the 

moment  of  inertia,  must  exert  a  perpendicular    effort   at   the    end 

— 2 
C,  as   AC.      The  strain  thence  produced  in  the  direction  AB    356. 

will  be   opposed    by  an   equal   and   opposite  effort  from  the  gate 

BC.      These  two  forces  constitute  the   power  which  closes  the 

gates.       If  the  angle  ACB  be   very   acute,  the  gates  would   close 

feebly  ;    if  on  the  other   hand  ACB   be    too   obtuse,    the    gates 

would  occasion  a  great  strain    upon   the  sides  of   the   sluice.     This 

— 2 
strain  in  the  direction  CA  is  as  AC,  other  things   being  the  same  j 

it  is  also  as  the  width  of  the  canal  or  as  AD,  and  for  the   reason 

above  given,  inversely  as  CD ;  that  is,  the  force  in  question  is  as 

AC'x  AD 
CD       ' 

Mech.  39 


t 


306  Hydrostatics. 

2 

AC~ 

or,  the  width  of    the  canal  being  constant,   as    ^m'     ^^'^  have, 

therefore,  only  to  find  when  this  is  a  minimum.  It  is  evidently 
equal  to  the  diameter  of  the  circle  circumscribing  the  triangle 
Geom.  ACB  \  and  since  the  least  circumscribins:  circle  is  that  whose  cen- 
tre  is  D,  we  infer  that  llie  angle  ACB  of  the  meeting  of  the  gates 
should  be  a  right  angle. 

423.  Water  is  sometimes  conveyed  in  pipes,  which,  according 
to  what  is  above  said,  must  sustain  a  pressure  in  proportion 
to  the  height  of  the  source.  There  is  moreover  a  force  exerted 
pj  2J0  upon  the  interior  of  the  pipe  depending  upon  its  diameter.  ADE 
being  a  transverse  section  of  the  pipe,  let  t^  be  a  particle  of  the 
fluid  pressed  by  two  contiguous  particles  B,  C,  and  kept  in 
equilibrium  by  the  resistance  or  tenacity  of  the  substance  of 
48.  the  pipe.  These  three  lorces  will  be  represented  by  the  three 
sides  of  a  triangle  formed  upon  their  directions,  or  by  the  three 
sides  of  the  similar  triangle  OBC,  formed  by  lines  drawn 
perpendicularly  to  the  directions  of  the  forces.  Now  if  we  sup- 
pose BC  io  become  indefinitely  small,  or  that  the  forces  exerted 
by  B,  C,  are  directly  opposed  to  each  other,  each  will  be  repre- 
sented by  the  radius  of  the  section.  Thus,  the  height  of  the  source 
being  the  same,  and  the  substance  of  the  pipe  the  same,  its  thick- 
ness ought  to  be  in  proportion  to  the  radius  or  diameter  of  the 
bore. 

We  may  apply  the  same  conclusion  to  cylindrical  vessels  gener- 
ally destined  to  hold  fluids.  Large  casks  are  required  to  be  made 
stronger  than  small  ones,  in  the  compound  ratio  of  their  diameter 
and  height.  The  same  precaution  is  to  be  observed,  moreover, 
with  regard  to  steam-pipes  and  steam-boilers  of  different  dimen- 
sions, for  the  above  reasoning  will  hold  true  equally  in  the  case  of 
elastic  fluids,  as  in  that  of  liquids. 


Solids  immersed  in  Fluids, 

424.  Since  the   efforts  which    a   fluid    makes   in  a    horizontal 
direction   mutually   destroy   each   other,    in    order   to   preserve   a 


Solids  immersed  in  Fluids.  307 

body  in  any  given  position  in  a  fluid,  it  is  only  necessary 
to  destroy  the  vertical  part  of  the  pressure ;  and  to  effect  this, 
tivo  things  are  required,  namely,  (1.)  A  downward  effort 
equal  to  that  of  the  upward  pressure  of  the  fluid  ;  and  (2.) 
A  coincidence  of  these  efforts  in  the  same  vertical  line.  Now 
the  vertical  upward  pressure  or  buoyancy  of  the  fluid  is  equal 
to  the  weight  of  the  portion  of  fluid  displaced  ;  hence,  if  the  por- 
tion of  fluid  displaced  iveighs  more  than  the  immersed  body,  the 
body  will  float,  and  it  ivill  elevate  itself  until  the  portion  of  fluid 
answering  to  the  part  immersed,  shall  iveigh  just  as  much  as  the 
entire  body. 

Accordingly,  if,  when  a  body  floats,  we  add  to  it  a  certain 
weight,  or  take  a  certain  weight  from  it,  it  will  sink  or  rise  until 
the  increase  or  diminution  of  the  fluid  displaced  shall  become 
equal  to  this  new  weight.  If  the  weight  added  or  subtracted 
be  small  conqjared  with  that  of  the  body  itself,  the  quantity  IK, 
by  which  the  section  AB  is  depressed  or  elevated,  will  be  soFig.207. 
much  the  less,  according  to  the  smallness  of  this  new  weight 
compared  with  the  extent  of  the  section  AB.  When,  therefore, 
this  new  weight  is  inconsiderable,  and  the  section  JIB  is  large, 
AB  and  A'B^  may  be  considered  as  equal,  and  the  difference 
in  the  bulk  of  fluid  displaced,  occasioned  by  the  supposed 
change  of  weight,  may  be  estimated  by  the  surface  .^i?  multiplied 
by  IK,  that  is,  by  AB  X  IK.  Therefore  if  iv  represent  the 
weight  of  a  cubic  foot  of  the  fluid,*  lo  X  AB  X  IK  will  express 
the  weight  of  the  bulk  in  question,  the  surface  ./^l?,  and  the  alti- 
tude /^  being  estimated  in  feet.  Thus,  \^  iv'  be  the  weight  added 
or  subtracted,  we  shall  have 

w  X  AB  X  IK  =  w', 

from  which  we  deduce 

IK—  ^'        ' 

^^  —  w  X  AB' 

that  is,  in  the  case  of  a  vessel,  for  example,  in  order  to  find  hoio 
much  a  certain  addition  to  the  cargo  ivill  sink  the  vessel,   we  divide 

*  This  in  the  case  of  fresh  water  is  at  a  mean  very   nearly 
G2^"'-,  and  in  that  of  sea-water  G4"'-. 


308  Hydrostatics. 

the  value  of  this  addition  zt/,  hy  the  surface  of  the  section  made 
at  the  water's  edge  (estimated  in  square  feet),  muhiplied  by  the 
weight  of  a  cubic  foot  of  water. 

When  a  vessel  is  sheathed,  the  bulk  of  the  hull  is  aug- 
mented by  a  quantity,  the  method  of  calculating  which  has 
Cal.  122.  been  made  known.  The  effect,  therefore,  is  the  same,  as  if  the 
cargo  were  diminished  by  the  following  quantity,  namely,  the 
weight  of  a  mass  of  water  equal  to  the  augmentation  of  the  hull, 
minus  the  weight  of  the  materials  which  compose  the  sheathing. 
Accordingly,  it  may  easily  be  determined  how  much  less  or  more 
the  vessel  would  sink. 

425.  The  weight  of  a  body  remaining  the  same,  its  bulk  may 
be  enlarged  at  pleasure,  by  forming  it  so  as  to  inclose  a  space. 
There  is  no  substance  therefore  so  heavy  that  it  may  not  be  made 
to  float. 

426.  Since,  when  the  weight  of  a  body  is  diminished  with- 
out changing  its  bulk,  it  must  elevate  itself  with  an  effort  which 
can  be  counterbalanced  only  by  a  weight  equal  to  that  which 
has  been  taken  from  it,  it  will  be  seen  that  this  upward  pressure 
of  water  may  be  advantageously  employed  in  raising  large  mass- 
es; in  drawing  up  vessels,  for  example,  from  the  bottom  of  bays 
and  rivers,  by  attaching  them  to  other  vessels,  floating  above,  and 
deeply  laden  with  stones  or  water,  afterward  to  be  thrown 
overboard. 


Specific  Gravities. 

427.  In  general,  if  S  be  the  specific  gravity  of  a  floating 
body,  or,  which  is  the  same  thing,  the  weight  in  ounces  of  a 
cubic  foot  of  this  body,*  the  matter  being  supposed  to  be  uni- 
formly distributed,  6  its  bulk,  S'  the   specific  gravity  of  the   fluid, 

*  Water  is  the  unit  to  which  we  refer  all  substances,  except  the 
gases,  in  estimating  their  specific  gravities.  It  is  immaterial 
what  bulk  be  used  for  this  purpose,  or  whether  any  particular 
bulk,  provided  the  two  in  question  be  equal.     A   cubic  foot  of  any 


Specific  Gravities.  300 

and  b'  the  bulk  of  the  part  immersed,  the  weight  of  this  body- 
will  he  S  b  ;  and  that  of  the  fluid  displaced  will  be  S^  b' ;  whence 
we  have 

Sb     =     S'     b'y 


an  equation  which  gives 


^—  S'  ■> 


from  which  it  will  be  seen,  that  the  weight  S  b  oi  the  body  remain- 
ing the  same,  the  part  immersed  will  always  be  so  much  the  less, 
according  as  the  specific  gravity  of  the  fluid  is  greater. 

Moreover,  this  same  equation  is  equivalent  to  the  proportion, 

b   :  b'  -.'.  S'  :  S', 

that  is,  the  bulk  of  the  body  is  to  that  of  the  part  immersed,  inverse- 
ly as  the  specific  gravity  of  the  body  to  that  of  the  fluid. 

428.  If  the  immersed  body  weigh  more  than  an  equal  bulk 
of  the  fluid,  it  must  sink  ;  and  it  can  be  retained  only  by  a  force 
equal  to  the  excess  of  its  weight  above  that  of  an  equal  bulk  of 
the  fluid.     Now  if  we   represent    the  specific  gravity  of  the  fluid, 

substance  compared  in  point  of  weight  with  the  same  bulk  of 
water  would  evidently  give  the  same  result  as  any  other  measure  ; 
and  correct  results  once  obtained,  we  can  substitute  such  other 
bulks  as  we  choose.  The  cubic  foot  has  this  particular  advantage, 
that,  if  the  measures  be  taken  at  the  temperature  of  50°  of  Fahren- 
heit, the  same  numbers  which  express  the  specific  gravities,  the 
decimal  point  being  removed  three  places,  or  the  whole  being 
multiplied  by  1000,  give  the  absolute  weight  in  avoirdupois  ounces  ; 
since  a  cubic  foot  of  water  at  this  temperature  weighs  1000  ounces. 
It  is  usual,  however,  in  determining  specific  gravities,  to  refer  to 
the  temperature  of  60°,  at  which  a  cubic  foot  of  water  weighs 
62,353'^-,  or  a  little  less  than  1000  ounces.  In  the  more  accurate 
experiments,  however,  upon  this  subject,  the  absolute  weight  is  first 
ascertained,  and  the  cubic  inch  is  taken  as  the  measure,  this  bulk 
of  distilled  water  at  the  temperature  of  60°,  being  estimated  at 
252,525  grains.  In  France  the  temperature  preferred  is  that  of  the 
maximum  density  of  water,  or  39°, 39.  The  atmosphere  is  em- 
ployed as  the  unit  in  estimating  the  specific  gravities  in  gases. 


310  Hydrostatics. 

and  that  of  the  body  immersed  in  it,  by  S',  S,  respectively,  and 
the  bulk  of  the  body  by  J,  we  shall  have,  S  b  —  ^S'  b  for  the 
excess  of  the  weight  of  the  body  above  that  of  an  equal  bulk  of 
the  fluid.  Hence,  if  we  suppose  that  this  body  is  retained  by 
means  of  a  thread,  attached  to  the  beam  of  a  balance,  and  that 
w  is  the  weight  with  which  it  is  kept  in  equilibrium  ;  we  shall 
have 


IV 


=  Sb  —  S'  b. 


from  which  we  obtain, 


S  Sb 


S'  ~  S  b 


Now  S  b  is  the  absolute  weight  of  the  body,  and  w  its  weight  in  the 
fluid ;  knoiving,  therefore,  the  absolute  iveight  of  a  body,  and  its 
weight  when  immersed  in  a  fluid,  we  easily  determine  the  ratio  of 
their  specific  gravities,  by  dividing  by  the  absolute  weight  of  the 
body  the  difference  of  these  two  weights.  If,  for  exa«nple,  the  ab- 
solute weight  of  a  body  be  6  ounces,  and  its  weight  when  immersed 
in  water  5  ounces,  we  divide  the  absolute  weight  6  by  the  diffe- 
rence, and  the  quotient  f  shows  that  the  specific  gravity  of  the  body 
is  6  or  is  to  that  of  the  fluid  as  C  is  to  1. 

In  other  words,  since  the  loss  of  weight  sustained  by  a  body 
on  being  immersed  in  a  fluid  is  equal  to  the  weight  of  its  own  bulk 
of  that  fluid,  by  proceeding  as  above,  we  shall  have  the  weight  of 
equal  bulks  of  the  substances  in  question,  and  consequently  their 
relative  weights  or  specific  gravities. 

If  the  solid  substance  whose  specific  gravity  is  sought  be  lighter 
than  water  with  which  it  is  to  be  compared,  it  may  be  made  to  sink 
by  attaching  to  it  a  heavier  body,  whose  water-weight  is  known  ; 
then,  by  subtracting  this  from  the  water-weight  of  the  compound 
body,  we  shall  have  that  of  the  body  in  question. 

When  the  substance  to  be  weighed  is  soluble  in  water,  as  com- 
mon salt,  for  instance,  it  may  be  covered  with  a  thin  coat  of  melted 
wax,  which  is  very  nearly,  and  may  be  made  exactly  of  the  same 
specific  gravity  with  water.  The  body  will  thus  be  protected,  and 
the  loss  of  weight,  on  being  immersed,  will  be  the  same  as  if  water 
occupied  the  place  of  the  covering. 


Specific  Gravities.  311 

Another  method  is  to  determine  the  specific  gravity  of  the  body 
with  reference  to  some  liquid,  as  alcohol  or  oil,  in  which  no  solu- 
tion takes  place,  and  whose  specific  gravity,  compared  with  water,  is 
known.  We  have  then  simply  to  use  the  proportion,  as  the  spe- 
cific gravity  of  water,  is  to  that  of  the  fluid  used  so  is  the  result 
above  found  to  the  result  sought. 

429.  If  the  same  body  be  immersed  in  another  fluid,  whose 
specific  gravity  is  denoted  by  S^',  and  w'  represent  the  weight 
necessary  to  counterbalance  it ;  as  in  the  former  case  we  had 
w  :=  S  b  —  S'  b,\\e  shall  have,  in  like  manner,  it/  z=z  S  b  —  S"  b. 
Now  these  two  equations  give 

S'b  =  Sb  —  w,     and     S'-'b  =  Sb  —  w'; 

whence,  dividing  this  last  by  the  preceding,  we  obtain 
S"        Sb  —  7v' 


S'  ~  Sb  —  w 

Knowing,  therefore,  the   absolute  weight   S  6   of  a  body,    and  its 

weight  w'  in  a  fluid,  and  its   weight  iv  in  any  other  fluid,  we  easily 

S" 
determine   the   ratio    -^   of    the   specific   gravities  of  these   two 

fluids. 

By  taking  a  solid  glass  ball  and  grinding  it  to  such  a  size  that 
it  shall  lose  just  a  thousand  grains,  for  instance,  when  weighed  in 
distilled  water,  at  the  assumed  temperature,  then  by  observing  the 
loss  of  weight  I  in  grains,  sustained  on  being  weighed  in  any  other 
fluid,  we  shall  have 

I 


1000  :  I  ::  I   :  S  = 


1000' 


Thus  the  specific  gravity  of  the  fluid  in  question  is  obtafeed  by 
dividing  /  by  1000,  or  removing  the  decimal  point  three  places  to 
the  left.  If  a  larger  number  of  decimal  places  were  required,  we 
sh^ld  employ  a  ball  weighing  10000  or  100000  grains,  and  di- 
vide accordingly. 

430.  The  precious  metals  being  heavier  than  those  of  less 
value,  when  they  are  debased,  it  must  be  by  means  of  some  sub- 
stance of  less  specific  gravity,  and  the  compound  will  conse- 
quently   be   lighter   than   the   metal    it   is   intended   to   represent. 


312  Hydrostatics. 

Spirits,  on  the  other  hand,  are  lighter  than  the  liquids  with 
which  they  are  adulterated,  so  that,  in  each  case,  the  specific 
gravity  becomes  the  test  of  purity.  We  have,  hence,  a  curious 
and  important  problem,  namely,  knowing  the  specific  gravity  of 
the  two  ingredients  that  compose  a  compound,  and  the  specific 
gravity  of  the  compound,  to  find  the  proportion  of  the  ingredi- 
ents. This  proportion  may  be  estimated  by  weight,  as  is  usually 
the  case  with  respect  to  metals,  or  by  measure,  which  is  the  com- 
mon method  where  liquids  are  concerned.  The  weight  of  the  in- 
gredients being  sought,  we  put  x  for  that  of  the  denser,  and  y  for 
that  of  the  rarer  or  lighter,  c  representing  that  of  ihe  compound, 
and  S,  S',  S",  denoting  their  specific  gravities  respectively.  We 
have  X  -\-  y  :=  c,  also,  on  the  supposition  that  the  bulk  of  the 
compound  is  equal  to  that  of  the  constituent  parts,  the  specific 
gravity  would  be  the  same  whether  the  compound  were  consid- 
ered as  one  entire  mass,  or  as  composed  of  two  distinct  parts,  and 
the  weight  divided  by  the  specific  gravity  in  the  one  case  would 
be  equal  to  the  weight  divided  by  the  specific  gravity  in  the 
other,  that  is, 

S"^  S'  ~  S"' 

To  find  X,  we  substitute  in  this  equation  the  value  of  y  deduced 
from  the  first,  namely,  y  ^=  c  —  x,  which  gives 


or  ^y 


whence 


a;        c  —  X  c 


X  jS'  —  S)  _  ±_  S_c_  _  jS'  —  S^ 

SS'        ~  S"       SS'  ~      S'  S"      ^' 


_       SS'  jS'—S")     _  {S'  —  S")S     _  (S"  —  S')  s 

*  -  {s—sj  ^    s' n"    ^-  {S'  —  S}S"  ^  -  (szirs')^^' 


In  like  manner,  we  have 


_  {S—S")S' 

y  —  (s— s')  s"  ^' 


1:1 


:C>'-  f^'C^te^-i^ 


Specific  Gravities.  313 

Suppose  a  compound  of  gold  and  silver  to  have  a  specific  grav- 
ity equal  to  14,  that  of  pure  gold  being  19,3,  and  that  of  pure  sil- 
ver 10,5  J  we  have 

S  =  19,3,     S'  =  10,5,     S''  =  14 ; 

and  consequently, 

S"  —  S'  =  3,5, 
S  —S'  z=z  8,8, 
5f   —S''=  5,3 ; 

whence,  by  the  formula,  the  proportional  weight  of  gold  will  be 

3,5  X  19,3  _  67,55  _  ^  .r 
8,8  X  14    ~  123,2  —  "'^''* 

The  formula  for  the  value  of  y  gives  for  the  proportional  weight  of 
silver 

5,3  X  10,5  _  55,65  __ 
8,8  X  14    ~"  123,3  —     '     ' 

whatever  c  may  be  in  each  case. 

431.  Where  the  proportion  of  the  ingredients  by  bulk  or  measure, 
instead  of  by  weight,  is  sought,  as  in  the  case  of  liquids,  the 
process  is  shorter.  Calling  b,  b',  the  bulks  of  the  ingredient 
corresponding  to  the  specific  gravities  S,  S',  as  the  weight  is 
equal  to  the  bulk  multiplied  by  the  specific  gravity,  and  the  weight 
of  the  compound  is  equal  to  the  sum  of  the  weights  of  the  ingre- 
dients, we  have 

S"  X  ib  +  b')     or     S"  b  +  S"b'  =  Sb-{-  S'  b', 


whence 


or 


that  is. 


S"b  —  Sb  =  S'b'  —  S"  b', 
h  [S"  ^  S)  =  b'  {S'  ^  S") ; 
b  :  b'  ::  S'  —  S"  :  S"  —  S. 


Suppose   a  certain   spirituous  liquor  to  have   a  specific  gravity 
equal  to  0,93,  that  of  higlily  rectified  alcohol  or  pure  spirit  being 
Mech.  40 


314  Hydrostatics. 

0,83  and  water  1  ;  we  have  in  this  case 

S  =  0,83,     S'  —  I,     and     S"  =  0,93, 

which  gives, 

S'  —  S"  =  0,07,     S"  —  S  =  0,1, 

and  by  substitution, 

h  :  h'  ::  0,07  :  0,10; 

that  is,  the  proportion  by  measure  of  pure  spirit  to  that  of  water, 
is  as  7  to  10.  In  what  is  called  proof  spirit,  it  is  required  that  the 
constituent  parts  of  water  and  spirit  should  be  equal,  which  gives  a 
specific  gravity  of  0,925,  and  the  degree  above  or  below  proof  is 
usually  denoted  by  tlie  number  of  gallons  of  water  to  be  added  to 
or  taken  from  100  gallons  of  the  liquor  in  question,  to  bring  it  to 
the  required  standard  or  proof. 

But  more  expeditious  methods  have  been  devised  for  deter- 
mining the  proportion  of  alcohol  in  spirits.  It  is  evident  that  if 
a  great  variety  of  mixtures  of  known  proportions  were  prepared, 
and  hollow  glass  balls  or  heads  were  so  ada|)ted  to  each  in  re- 
spect to  specific  gravity,  as  just  to  remain  suspended  in  any  part 
of  the  fluid  ;  by  marking  the  known  proportion  of  alcohol  and 
spirits  in  each  case  on  the  respective  balls,  similar  unknown 
mixtures  might  be  readily  examined  as  to  their  proportion  of 
alcohol.  It  would  only  be  necessary  to  make  trial  of  different 
balls  until  one  was  found  which  would  tend  neither  to  rise  nor 
sink  when  immersed  in  the  fluid.  But  instead  of  a  large  number 
of  such  balls,  a  single  one  with  a  long  graduated  stem  and  mov- 
able weights,  as  represented  in  figure  21 1,  might  be  employed  to 
the  same  purpose.  The  ball  itself  should  be  of  such  a  specific 
gravity  as  just  to  sink  to  the  commencement  of  the  graduation  on 
the  stem  when  suspended  in  alcohol,  and  the  weights  to  be  at- 
tached should  be  such  as  to  cause  the  ball  and  some  part  of  the 
stem  to  sink  in  any  mixture  of  less  specific  gravity  than  pure 
water.  Then  the  weight  together  with  the  divisions  of  the  stem, 
which  serve  to  subdivide  the  difference  between  two  weights, 
would  indicate,  like  the  separate  balls,  the  specific  gravity   of  the 


Specific  Gravities.  315 

liquid  and  consequently  its  proportion  of  alcohol.      An  instrument 
so  constructed,  is  called  a  hydrometer.* 

432.  In  making  use  of  the  hydrometer,  and  in  experiments 
generally  upon  specific  gravities,  there  are  several  particulars  to  be 
taken  into  consideration. 

(1.)  Since  all  bodies  expand  vi'ith  heat  and  contract  with 
cold,  it  will  be  perceived  that  the  specific  gravity  of  bodies  is 
modified  by  temperature.  Thus  bodies  are  specifically  heavier 
in  winter  than  in  summer,  and  the  same  spirit  would  indicate  dif- 
ferent proportions  of  alcohol  at  different  seasons,  regard  not  be- 
ing had  to  this  circumstance.  Accordingly  a  thermometer  is  a 
necessary  appendage  to  a  hydrometer,  and  the  correction  for 
temperature  is  applied  by  means  of  a  movable  scale,  containing 
the  degrees  of  the  thermometer,  sliding  upon  another  scale  on 
which  are  placed  the  numbers  of  the  several  weights,  including  the 
stem. 

(2.)  When  two  substances  are  mixed  together,  there  is  often 
a  mutual  penetration  of  parts,  whereby  the  specific  gravity  is 
increased,  and  there  would  seem,  by  the  foregoing  methods,  to  be 
a  greater  proportion  of  the  heavier  ingredient  than  actually  ex- 
ists. Thus  a  pint  of  water  and  a  pint  of  alcohol  do  not  make  a 
quart  of  liquid.  The  defect  is  sometimes  a  40th  part.  On  the 
other  hand,  the  bulk  in  certain  cases  is  augmented  by  compound- 
ing. A  cubic  inch  of  tin  mixed  with  a  cubic  inch  of  lead  will  make 
a  compound  exceeding  two  cubic  inches.  No  accurate  allowance 
can  be  made  for  such  changes ;  consequently  the  methods  above 
given,  become  in  a  degree  defective,  and  the  results  to  a  certain 
extent  uncertain. 

(3.)  The  number  of  ingredients  may  exceed  two,  or  the  na- 
ture of  one  or  more  of  the  ingredients  may  be  unknown,  hi  all 
such  cases,  the  problem  is  in  its  nature  indeterminate. 

433.  Questions  sometimes  occur,  especially  in  chemical  re- 
searches, the  reverse  of  those  we  have  been  considering,  namely, 

*  There  is  a  great  variety  of  instruments  of  this  kind,  all  de- 
pending on  the  same  general  principles.  They  are  known  also  by 
a  variety  of  names,  as  areometer,  gravimeter,  alcoholometer,  i)cse- 
liqueiir,  essay-instrument ,  &c. 


316  Hydrostatics. 

to  find  the  specific  gravity  of  a  compound,  by  means  of  that  of 
each  of  the  ingredients ;  and  the  rule  which  has  generally  been 
given,  is  to  take  the  arithmetical  mean  of  the  specific  gravities  of 
the  ingredients. 

It  will  be  observed  that,  the  greater  the  weight  of  a  body," 
other  things  being  the  same,  the  greater  the  specific  gravity  ;  also 
that  the  less  the  bulk,  other  things  being  the  same,  the  greater 
the  specific  gravity  ;  that  is,  the  specific  gravity  of  one  body  is  to 
that  of  another,  as  the  weight  of  the  first  divided  by  its  bulk,  is  to 
the  weight  of  the  second  divided  by  its  bulk,  and  hence  the  mean 
specific  gravity  of  the  two  will  be  found  by  dividing  the  sum  of  the 
weights  by  the  sum  of  the  bulks.     Thus, 


and 

from  which  we  obtain. 

7>         ^^ 

and 

also. 


,7/  7P  W'   7V  S'  -\-  W'  S 


Hence,  calling  M  the  mean  specific  gravity  sought,  from  the 
equat'ion  M  ;=  ,  7]  ,,,  above  shown  to  be  true,  by  general  reason- 
ing, we  obtain,  by  substitution, 

^r-        ^S'  +  w'S 

Let  gold,  for  example,  of  a  specific  gravity  19,36,  be  alloyed 
in  equal  weights  w-ith  copper  of  a  specific  gravity  8,87,  we  shall 
have 

M-  (1 +  1)19,36' 8,87  _  2x  171,7232  _,o,^. 
8,87  +  19,36         ~  28,23  —  a^j^"? 

whereas  the  arithmetical  mean  is 

15£l±^  =  I4,li. 


Specific  Gravities.  317 

434.  It  will  be  recollected  that  the  specific  gravity  of  a  solid- 
body  is  determined  by  dividing  its  absolute  weight  by  the  loss 
sustained  on  its  being  weighed  in  pure  water,  and  the  specific 
gravity  of  a  fluid,  whether  liquid  or  gas,  by  comparing  the  loss  of 
weight  sustained  by  the  same  solid  when  weighed  in  pure  water 
with  that  sustained  on  its  being  weighed  in  the  fluid  in  question. 
According  to  this  method,  therefore,  the  absolute  weight  of  a 
body  is  necessary  in  both  cases.  But  it  is  diflicult  to  obtain  this 
unaffected  by  the  fluid  of  the  atmosphere  in  which  we  are  im- 
mersed. The  usual  operation  of  weighing,  except  where  the 
weight  itself  and  the  thing  weighed  happen  to  be  equal  in  bulk, 
must,  from  what  has  been  said,  be  more  or  less  incorrect.  But 
the  specific  gravity  of  the  atmosphere  being  ascertained,  on  the 
supposition  that  it  is  always  the  same,  or  such  as  to  admit  of  its 
changes  of  density  being  determined  at  all  times,  allowance  may 
be  made  for  its  effect  on  the  weight  of  bodies,  more  especially 
as  it  is  an  exceedingly  light  fluid,  and  scarcely  requires  to  be 
noticed,  except  in  very  nice  experiments,  or  where  the  bulks  of 
bodies  are  very  considerable.  The  best  way  of  determining  the 
specific  gravity  of  the  atmosphere,  and  of  gases  generally,  is  to 
weigh  directly  a  vessel  of  known  dimensions,  when  empty,  and 
again  when  filled  with  the  fluid  in  question.*  It  is  thus  found 
thai  a  vessel  of  three  hundred  cubic  inches,  for  example,  weighs 
92,4  grains  more  when  filled    with  air  in  its  ordinary  state,  than 

*  The  vessel  should  be  of  considerable  size,  that  is,  sufficient 
to  contain  at  least  three  or  four  hundred  cubic  inches.  It  might 
be  of  a  globular  shape,  as  represented  in  figure  212,  with  a  nar- 
row neck  and  nicely  fitted  stop-cock  C.  Its  capacity  would  be  best 
ascertained  by  filling  it  accurately  with  mercury,  and  then  pouring 
the  liquid  into  a  prismatic  vessel,  which  might  be  easily  measured. 
The  air  might  be  expelled  also  in  the  same  way,  by  connecting 
with  the  neck  AB  a  tube  of  about  three  feet  in  length,  and  suffer- 
ing the  mercury  to  discharge  itself  through  this  tube,  held  upright 
with  its  lower  end  immersed  in  the  same  liquid.  When  the  mercu- 
ry had  left  the  ball,  upon  turning  the  stop-cock  we  should  efTectual- 
ly  exclude  the  air;  but  there  would  remain  a  small  portion  of  the 
vapor  of  mercury.  Ordinarily  the  air  is  exhausted  by  means  of 
an  air-pump,  and,  although  the  air  cannot  thus  be  wholly  with- 
drawn, the  small  proportion  which  is  left  may  be  measured  and 
allowance  made  accordingly,  as  will  be  shown  hereafter. 


,318  Hydrostatics. 

it  does  when  empty.  This  divided  by  300,  or  ||-'4>  gi'^^s  for  the 
absolute  weight  of  a  cubic  inch  of  air  0,308  parts  of  a  grain. 
By  dividing  this  by  25i,525,  the  weight  in  grains  of  a  cubic  inch 
of  water,  gives  0,00122  or  j^-^  nearly  for  the  specific  gravity  of 
common  air,  at  the  surface  of  the  earth  in  its  mean  state  of  den- 
sity and  moisture,  the  temperature  being  tliat  to  which  specific 
gravities  are  generall}  referred  by  English  philosophers,  name- 
ly, 60°  of  Fahrenheit.  Hence  bodies  weighed  in  air  lose  at  a 
mean  ji^  part  of  the  weight  lost  on  being  weighed  in  water. 
Accordingly,  if  we  weigh  a  body  in  air  and  increase  this  weight 
by  8^0  of  the  difference  between  the  air  and  water  weight,  we 
shall  hav(!  the  absolute  weight  very  nearly  ;  that  is,  if  w  be  the 
absolute  weight,  tv'  the  air  weight,  and  w''  the  water  weight,  we 
shall  have  iv  =  w'  -\-  j^^  {w'  —  w"^  very  nearly.* 

If  it  were  proposed  to  find  how  much  a  solid  ball  must  weigh  in 
the  air  in  order  that  its  absolute  weight  may  be  1000  grains,  from 
the  equation 

1000  =  w'  +  ^1^  {w'  —  w'% 

we  should  have 

v/  =  1000  —  3!^  {w'  —  w"), 

or, 

y/  _  w"  being  328,  for  example,  w'  =  1000  —  0,4  =  999,6. 

*  To  be  strictly  correct,  the  formula  should  be 

but  the  object  of  this  formula  is  to  find  w,  and  when  we  have 
obtained  it  nearly,  we  may  substitute  this  value,  and  thus  ap- 
proximate the  true  value  of  to  to  any  degree  of  exactness.  But 
generally  speaking,  the  correction  derived  from  the  second  approx- 
imation is  very  small.  Thus,  in  the  example  that  follows  above, 
TJo"  ("''  —  ^")  '^  ^'^  grains,  and  the  second  approximation  would 
give  only  -5-^^  of  0,4  grains  or  0,0005  of  a  grain.  Where  the  re- 
sults are  intended  to  be  very  accurate,  the  coefficient  -g^^  should 
be  corrected  for  the  particular  state  of  the  atmosphere  at  the  time 
of  the  experiment,  the  method  of  doing  which  depends  upon  in- 
struments to  be  described  hereafter. 


Specific  Gravities.  319 

435.  According  to  what  is  above  laid  down,  the  specific  gravity    427. 
of  a  body  multiplied  by  its  bulk  gives  its  weight.     Now  the  density 
multiplied   by  the  bulk  gives  what  is  called  the  mass  of  the  body,      so. 
which    is   proportional  to  its  weight.     Hence  the  specific  gravity     272. 
multiplied  by  the  bulk,  is  proportional  to  the   density  multiplied  by 
the  bulk ;  therefore  the  specific  gravities  of  bodies  are  proportional 
to  their  densities.     Thus  S,  <S',   being  the  specific  gravities,  w,  n/, 
the  weights,  b,  b',  the  bulks,  A,  A',  the  densities,  and  m,  m'.  the 
masses  of  two  bodies, 

S  b  =  w,     and     S'  b'  =  w' ; 


also, 
<Vhence 


or, 


A  6  :=  m  :  A'  b'  =^  m'  :  :  w  :  w' ', 
A  b  :  A'  b'  :  :  S  b  :  S'  b', 
A    :    A'     ::     S    :     S'. 


By  employing  the  same  unit  in  both  cases,  as  water,  for  exam- 
ple, at  the  same  temperature,  the  specific  gravhies  would  be  equal 
to  the  densities. 

If  fluids  of  various  densities,  and  not  disposed  to  unite  by 
any  chemical  affinity,  be  poured  into  a  vessel,  they  will  arrange 
themselves  in  horizontal  strata,  according  to  their  respective 
densities,  the  heavier  always  occupying  the  lower  place.  This 
stratified  arrangement  of  the  several  fluids  will  succeed,  even 
though  a  mutual  attraction  should  subsist,  provided  only  that  its 
operation  be  feeble  and  slow.  Thus,  a  body  of  quicksilver  may 
be  at  the  bottom  of  a  glass  vessel,  above  it  a  layer  of  concen- 
trated sulphuric  acid,  next  this  a  layer  of  pure  water,  and  then 
anotlier  layer  of  alcohol.  The  sulphuric  acid  would  scarcely 
act  at  all  upon  the  mercury,  and  a  considerable  time  would 
elapse  before  the  water  sensibly  penetrated  the  acid,  or  the  al- 
cohol the  water.  Bodies  of  different  densities  might  remain 
suspended  in  these  strata.  Thus,  while  a  ball  of  platina  would 
lie  at  the  bottom  of  the  quicksilver,  an  iron  ball  would  float  on  its 
surface  ;  but  a  ball  of  brick  would  be  lifted  up  to  the  acid  and  a 
ball  of  beech  would  swim  in  the  water,  and  another  of  cork  might 
rest  on  the  top  of  the  alcohol. 


330 


Hydrostatics. 


Specific  Gravities  of  the  more  remarkable  Substances. 


Platinum,  purified, 

hammered, 

laminated,     .     . 

drawn  into  wire, 

Gold,  pure  and  cast,      .     . 

hammered,    . 

Mercury,    .... 
Lead,  cast,     .     . 
Silver,  pure  and  cast, 

hammered, 

Bismuth,  cast. 
Copper,  cast,    .     . 

wire,       .     . 

Brass,  cast, 

wire,    .... 

Cobalt  and  nickel,  cast. 
Iron,  cast,       .... 
Iron,  malleable,     . 
Steel,  soft,       .     . 
hammered. 


Tin,  cast,  .     .     .     . 
Zinc,  cast,  .     .     . 
Antimony,  cast, 
Molybdaiiium, 
Sul|)hate  of  barytes. 
Zircon  of  Ceylon, 
Oriental  ruby,      .     . 
Brazilian  ruby, 
Bohemi:in  garnet,     . 
Oriental  topaz, 
Diamond,  .     .     .     . 
Crude  manganese, 
Flint  Glass,     .     .     . 
Glass  of  St.  Gobin, 
Fluor  Spar,     .     .     . 
Parian  marble, 
Peruvian  emerald,    . 
Jasper,     .... 
Carbonate  of  lime,  . 
Rock  crystal,     .     . 

Flint, 

Sulphate  of  lime. 
Sulphate  of  soda,     . 
Common  salt,    . 
Native  sulphur,    .     . 
Nitre,      .... 
Alabaster,        .     .     . 
Phosphorus,      .     . 


19,50 

20,33 

22,07 

21,04 

19,2G 

19,36 

13,57 

11,35 

10,47 

10,51 

9,82 

.  8,79 

8,89 

.  8,40 

8,54 

.  7,81 

7,21 

.  7,79 

7,83 

.  7,84 

7,30 

.  7,20 

4,95 

.  4,74 

4,43 

4,41 

4,28 

.  4,53 

4,19 

.  4,01 

3,50 

.  3,53 

2,89 

.  2,49 

3,18 

.  2,34 

2,78 

.  2,70 

2,71 

,  2,65 

2,59 

2,32 

2,20 

2,13 

2,03 

2,00 

1,87 

1,77 


Plumbago, 1,86 

Alum, 1,72 

Asphaltum, 1,40 

Jet, 1,24 

Coal,  from  .  .  .  1,24  to  1,30 
Sulphuric  acid,      ....   1,84 

Nitric  acid, 1,22 

Muriatic  acid, 1,19 

Equal    parts     by    weight   of 
water  and  alcohol,     .     .     0,93 

Ice 0,92 

Strong  alcohol,  ....  0,82 
Sulphuric  aether,    ....  0,74 

Naphtha, 0,71 

Sea  water, 1,03 

Oil  of  sassafras,  ....     1,09 

Linseed  oil, 0,94 

Olive  oil, 0,91 

White  sugar, 1,61 

Gum  Arabic  and  honey     .     1,45 

Pitch, 1,15 

Isinglass, 1,11 

Yellow  amber,  .  .  .  .1,08 
Hen's  egg,  fresh  laid,    .     .     1,05 

Human  blood, 1,03 

Camphor,         0,99 

White  wax, 0,97 

Tallow, 0,94 

Pearl, 2,75 

Sheep's  bone,      ....     2,22 

Ivory, 1,92 

Ox's  horn, 1,84 

Lignum  vitre, 1,33 

Ebony,        1,18 

Mahogany,        1,06 

Dry  oak, 0,93 

Beech, 0,85 

Ash, 0,84 

Elm,  .  .  .  from  0,80  to  0.60 
Fir,        .     .     .    from  0,57  to  0,60 

Poplar, 0,38 

Cork 0,24 

Chlorine,  ....  0,00302 
Carbonic  acid  gas,  .  .  0,00164 
Oxygen  gas,  .  .  .  0,00134 
Atmospheric  air,  .  .  0,00122 
Azotic  gas,  .  .  .  0,00098 
Hydrogen  gas,    .     .     .     0,00008 


Spirit  Level.  321 

It  will  be  seen  by  the  foregoing  table,  that  there  are  gases 
much  lighter  than  the  atmosphere  ;  accordingly,  if  a  large  quan- 
tity of  one  of  these  fluids  were  confined  by  a  thin  covering,  as  oiled 
silk,  it  would  rise  in  the  atmosphere  as  a  cork  does  in  water  until 
it  reached  a  region  of  the  atmosphere  of  the  same  specific  gravity 
with  itself.  It  is  upon  this  principle  that  balloons  are  constructed. 
They  were  at  first  filled  with  air  rarefied  by  heat,  and  a  fire  was 
supported  in  a  car  placed  under  the  balloon  for  this  purpose.  Hy- 
drogen was  afterward  discovered,  and  proved  to  have  a  specific 
gravity  only  J^  of  that  of  common  atmospheric  air.  This  gas  is 
the  lightest  of  known  fluids,  and  will  consequently  require  a  less 
bulk  for  a  given  buoyancy  than  any  other.  It  is  universally  em- 
ployed in  the  construction  of  balloons. 


Spirit  Level. 

436.  This  instrument  consists  of  a  tube  AB,  nearly  filled  Fig. 213, 
with  alcohol  or  ether,  the  remaining  space  CD  being  occupied 
by  a  bubble  of  air.  Being  attached  to  the  vertical  or  horizon- 
tal circle  of  a  theodolite  or  other  instrument,  it  serves  to  deter- 
mine the  horizontal  position  of  a  line  or  plane,  since  if  either  end 
of  the  tube  is  higher  than  the  other,  the  bubble  of  air,  from  its 
specific  lightness,  will  indicate  it  by  tending  to  the  higher  part. 
It  is  usual,  where  great  exactness  is  required,  to  make  the  tube 
slightly  curved,  the  curvature  being  circular ;  the  bubble  will 
then  move  more  readily,  settle  itself  with  more  certainty,  and 
describe  equal  spaces  by  equal  changes  of  inclination.  A  good 
spirit  level  will  exhibit  a  movement  of  more  than  half  an  inch  for 
each  minute  of  inclination,  and  the  bubble  will  sensibly  alter  its 
position  by  a  change  of  five  seconds  in  the  inclination.  In  such 
a  tube,  the  radius  of  curvature  will  be  about  150  feet.  But  lev- 
els have  been  rendered  still  more  delicate.  Lalande  speaks  of 
one,  filled  with  ether,  the  bubble  of  which  passed  over  fourteen 
inches  by  equal  spaces  of  one  tenth  of  an  inch  for  every  second. 
The  radius  of  curvature  was  consequently  1719  feet,  or  nearly  one 
third  of  a  mile.* 

•  The  number  of  seconds  in  a  circle  is 

360  X  60  X  60  =  1296000. 
Mech.  41 


322  Hydrostatics. 


Of  the  Equilibrium  of  Floating  Bodies. 

437.  In  order  that  a  heavy  body  may  be  in  equilibrium  on 
the  surface  of  a  tranquil  fluid,  it  is  necessary  that  its  weight 
should  be  less  than  that  of  an  equal  bulk  of  the  fluid.  There 
is  an  exception  to  this  rule,  however,  in  the  case  of  very  small 
bodies,  the  particles  of  which  are  of  a  nature  not  to  exert  any 
attraction  upon  the  particles  of  the  fluid,  or  whose  attraction  is 
much  less  than  that  of  the  particles  of  fluid  for  each  other.  In 
this  case  the  fluid  is  depressed  below  its  level,  forming  a  little 
hollow  about  the  floating  body,  which  may  be  regarded  as  mak- 
ing a  part  of  the  bulk  of  the  body  ;  and  on  account  of  this  aug- 
mentation, it  is  evident  that  the  body  may  float,  although  its  spe- 
cific gravity  should  be  greater  than  that  of  the  fluid.  It  is  on 
this  account  that  a  needle  smeared  with  tallow  may  be  made  to 
float  on  water.  Small  globules  of  mercury  also  are  supported  in 
a  similar  manner.  To  render  this  effect  of  no  avail,  we  have 
only  to  suppose  the  floating  bodies  so  large,  that  the  void  space 
which  may  be  formed  about  them  (and  wiiich  is  always  very 
small)  may  be  neglected  when  taken  in  connection  with  a  bulk 
so  much  greater. 

438.  The  weight  of  a  body  being  smaller  than  that  of  an 
equal  bulk  of  the  fluid,  the  body  will  sink  till  the  weight  of  the 
fluid  displaced  becomes  equal  to  that  of  the  body  ;  and  when 
these  two  weights  are  thus  equal,  the  body  will  be  in  equilibri- 
um, if   its  centre  of  gravity   and  that  of   the  fluid   displaced,  or 

32.  the  centre  of  buoyancy,  are  situated  in  the  same  vertical.  With 
respect  to  homogeneous  bodies,  the  centre  of  buoyancy  coincides 
with  that  of  the  immersed  part  of  the  body  ;  and  that  the  weight 
of  the  fluid  displaced  may  be  equal  to  that  of  the  body,  it  is 
necessary   that  the  densities  should  be  in    the  inverse  ratio  of  the 

Calling  these  tenths  of  an  inch,  we  have,  by  dividing  by  10  and  by 
12, 10800  feet  for  the  absolute  length  of  the  circumference;  whence, 

2  n  or  6,2832  :  1  :  :  10800  :  1719. 

In  the  best  spirit  levels,  the  requisite  curvature  is  effected  and 
rendered  true  by  grinding  with  emery. 


Equilibrium  of  Floating  Bodies.  323 

bulks,  or  that  the  bulk  of  the  immersed  part  should  be  to  the 
entire  bulk  of  the  body,  as  its  density  is  to  that  of  the  fluid  ;  it  42" 
follows,  therefore,  that  the  determination  of  the  positions  of  equi- 
librium of  a  homogeneous  body,  placed  on  the  surface  of  a  fluid 
of  a  given  density  greater  than  that  of  the  body,  is  reduced  to  a 
problem  of  pure  geometry  which  may  be  very  simply  stated. 
It  is  required  to  cut  the  body  by  a  plane  in  such  a  manner,  that 
the  bulk  of  one  of  its  segments  shall  be  to  that  of  the  whole  body 
in  a  given  ratio,  and  that  the  centre  of  gravity  of  this  segment 
and  that  of  the  body  shall  be  situated  in  the  same  perpendicular 
to  the  cutting  plane. 

439.  There  are  different  kinds  of  equilibrium  depending  upon 
the  form  and  position  of  the  floating  body.  With  respect  to 
the  sphere,  for  example,  provided  its  density  be  less  than  that 
of  the  fluid,  it  will  remain  in  equilibrium  in  any  position  whatev- 
er, since  the  centre  of  gravity  and  that  of  buoyancy  continue  to 
be  in  the  same  vertical.  This  will  be  the  case,  also,  with  re- 
spect to  solids  of  revolution  generally,  on  the  supposition  that  the 
axis  remains  horizontal.  Such  an  equilibrium  is  called  an  equi- 
librium of  indifference.  But  when,  from  the  form  of  the  solid  or 
its  relative  density,  it  tends,  upon  being  inclined  a  little,  to  return 
to  its  position,  the  equilibrium  is  said  to  be  stable.  On  the  other 
hand,  if  its  tendency  after  a  slight  inclination  is  to  depart  from 
its  first  position,  the  equilibrium  is  denominated  unstable. 

440.  With  respect  to  the  different  positions  of  equilibrium  of 
the  same  solid,  there  is  a  remarkable  properly  which  may  be 
demonstrated  independently  of  any  calculation.  Let  us  suppose 
that  the  body  in  question  is  made  to  turn  about  a  movable  axis 
which  is  kept  constantly  parallel  to  a  fixed  and  horizontal  straight 
line,  and  that  it  is  made  to  pass  in  this  way  successively  through 
all  its  positions  of  equilibrium  in  which  the  axis  has  this  direc- 
tion ;  we  say  that  the  positions  of  stable  and  unstable  equilibrium 
will  succeed  each  other  alternately,  so  that  if  the  body  be  mov- 
ed from  a  position  of  stable  equilibrium,  the  next  position  will 
be  unstable,  the  third  stable,  and  so  on  till  it  has  returned  to  its 
first  position. 

Indeed,  while  the  body  is  yet  very  near  its  first  position,  it 
will  tend  to  return  to  it,  this  position  being  supposed  to  be  stable ; 


324  Hydrostatics. 

but  the  tendency  thus  to  return  will  gradually  diminish  as  it  re- 
volves, till  after  a  time  the  body  will  incline  the  other  way,  but  be- 
fore this  tendency  changes  its  sign  (to  borrow  an  expression  from 
algebra),  there  will  be  a  position  in  which  it  will  be  nothing, 
and  in  which  the  body  will  neither  incline  to  return  to  its 
first  position,  nor  to  depart  from  it  ;  this,  therefore,  will  be 
its  second  position  of  equilibrium.  Now  we  see  that  within  this 
part  of  its  revolution,  the  body  tends  to  return  to  its  first  posi- 
tion, and  consequently  to  depart  from  the  second.  Beyond  this 
point,  the  body  tends  to  depart  from  its  first  position,  and  at  the 
same  time  from  the  second ;  therefore  the  second  position  of 
equilibrium  is  not  stable,  since  on  each  side  of  it  the  body  tends 
to  depart  from  it.  Upon  its  passing  this  position,  its  tendency 
to  depart  from  it  diminishes  continually,  till  it  becomes  nothing ; 
and  beyond  this  the  body  tends  to  return  toward  its  second 
position.  The  point  where  this  tendency  is  nothing,  is  a  third 
position  of  equilibrium,  which  is  evldendy  stable ;  for  on  eacli 
side  of  it,  the  body  tends  to  return  to  it,  either  approaching  to- 
ward or  receding  from  its  second  position.  If  the  third  position 
is  stable,  it  may  be  shown  by  the  same  kind  of  reasoning  that 
the  fourth  is  not,  and  that  the  fifth  is,  and  so  on. 

Thus,  when  the  body  returns  to  its  first  position,  it  will  have 
necessarily  passed  through  an  even  number  of  positions  of  equi- 
librium, alternately  stable  and  unstable. 

441.  It  is  important  to  be  able  to  distinguish  a  stable  position 
of  equilibrium  in  a  floating  body  from  one  which  is  not  so.  In 
order  to  this  let  us  suppose  a  body  which  admits  of  being  divid- 
Fig.2I4.  ed  by  a  vertical  plane //i^/ into  two  parts  perfectly  similar,  both 
as  to  form  and  density.  Let  us  suppose,  moreover,  that  this 
body  is  made  to  depart  from  its  position  of  equilibrium,  in  such 
a  manner  that  this  section  HFl  remains  vertical,  and  that  after 
having  thus  disturbed  it,  we  leave  it  to  itself  without  impressing 
upon  it  any  velocity ;  m  this  way  the  section  HFl  will  remain 
in  the  same  vertical  plane,  during  the  whole  motion  of  the  body, 
for  the  two  portions  being  perfectly  similar  in  all  respects,  there 
is  no  reason  why  it  should  ever  depart  from  the  vertical  plane  in 
which  it  was  supposed  to  be  first  situated.  For  the  same  reason, 
the  centre   of  buoyancy  will   always  be  in  the  section  HFl,  as 


Equilibrium  of  Floating  Bodies.  325 

yypW  as  the  centre  of  gravity.  Let  G,  then,  be  the  centre  of 
gravity  ;  and  the  position  being  that  of  equilibrium,  let  B  be  the 
centre  of  buoyancy,  and  HI  the  intersection  of  the  level  of  the 
fluid  with  the  plane  HFI,  or  ihe  tvater-Une ;  in  this  position,  the 
straight  line  GB,  which  connects  the  two  centres,  is  vertical,  and 
consequently  perpendicular  to  the  straight  line  HI-  it  inclines 
generally  when  the  body  is  made  to  depart  from  this  position, 
and  at  the  same  time,  the  centre  of  buoyancy,  and  the  water-line, 
change  their  position  upon  the  plane  HFI.  I  will  suppose,  there- 
fore, that  this  centre  is  the  point  B',  and  this  line  the  straight 
line  HP,  when  the  equilibrium  has  been  disturbed ;  the  forces 
which  will  tend  to  put  the  body  in  motion  are  the  weight  of  the 
body  which  is  directed  according  to  the  virtical  GF^  drawn 
through  the  centre  of  gravity  G,  and  the  resultant  of  the  vertical 
pressures  of  the  fluid  upon  the  surface  of  the  body  ;  this  result- 
ant is  the  buoyancy  of  the  fluid,  and  is  equal  to  the  weight  of 
the  fluid  displaced,  and  is  exerted  at  the  point  B'  its  centre  of  136. 
gravity,  in  the  direction  contrary  to  that  of  gravity,  or  according 
to  the  vertical  B^Z.  This  vertical  and  the  inclined  straight  line 
GB  being  in  the  same  plane,  will  cut  each  other  in  a  certain 
point  M  called  the  metacentre.  It  is  on  the  position  of  this  point 
with  respect  to  the  centre  of  gravity  G,  that  the  stability  of  the 
equilibrium  depends.  The  point  M  may  be  taken  for  the  point  43. 
of  application  of  the  buoyancy  of  the  fluid,  which  will  then  be 
exerted  according  to  the  straight  line  MZ ;  the  body  will  there- 
fore be  acted  upon  by  two  parallel  and  contrary  forces,  applied 
at  the  extremities  of  the  straight  line  GAl.  It  is  now  proposed  to 
determine  in  what  direction  the  body  will  move,  and  whether 
these  forces  will  tend  to  restore  it  to  its  position  of  equilibrium,  or 
to  make  it  depart  further  from  this  position. 

442.  In  the  first  place,  if  they  be  unequal,  they  will  produce  a 
motion  of  oscillation  in  the  point  G.  For  the  centre  of  gravity  124. 
ought  to  move  just  as  if  the  two  forces  were  applied  directly  at  this 
point ;  therefore,  the  initial  velocity  being  nothing,  its  motion  will  be 
in  a  vertical  straight  line,  and  in  point  of  magnitude  equal  to  the 
excess  of  the  greater  of  the  two  forces  over  the  less.  If  at  the 
commencement  of  the  motion,  the  weight  of  the  body  exceeds  the 
buoyancy  of  the  fluid,  the  point  G  will  begin  to  descend  ;  its  mo- 
tion will  at  first  be  accelerated,  but  according  as  the  body  sinks 


326  Hydrostatics. 

in  the  fluid,  the  portion  displayed  will  be  greater  and  greater, 
and,  consequently  the  buoyant  effort  will  increase,  till  at  length 
it  wili  become  equal  to  the  weight  of  the  body.  The  point  G 
will  still  continue  to  move  on  in  the  same  direction  by  virtue  of 
its  acquired  velocity,  but  then  the  buoyancy  of  the  fluid  exceed- 
ing the  weight  of  the  body,  its  velocity  will  be  retarded  contin- 
ually, till  finally  the  downward  motion  of  G  will  cease,  and  then  it 
will  begin  to  return  toward  its  first  position,  and  thus  it  will  continue 
to  oscillate  till  the  motion  is  entirely  destroyed  by  the  resistance 
of  the  fluid.  The  extent  of  these  oscillations  will  be  smaller,  ac- 
cording to  the  difference  at  the  outset  between  the  weight  of  the 
body  and  that  of  the  fluid  displaced,  compared  with  the  weight 
of  the  body.  If  the  body  be  but  little  removed  frotn  its  position 
of  equilibrium,  this  difference  will  be  small,  the  extent  of  the 
oscillations  will  consequently  be  small,  and  will  not  materially  affect 
the  stability  of  the  body. 

443.  During  these  oscillations  of  the  point  G,  the  body  will 
turn  about  this  centre,  in  precisely  the  same  manner  as  if  it  were 
137.  fixed  ;  its  motion  of  rotation  will  be  produced,  therefore,  by  the 
buoyancy  of  the  fluid,  which  acts  at  the  point  M  according  to 
the  direction  MZ ;  and  its  state  of  equilibrium  will  be  stable  or 
unstable,  according  as  the  straight  line  GB  tends  to  approach  to, 
or  recede  from,  the  vertical.  Now  it  is  evident  from  inspection, 
that  the  buoyancy  of  the  fluid  will  tend  to  restore  the  straight 
line  GB  to  its  vertical  position,  whenever  the  point  M  is  above 
the  point  G  ;  on  the  other  hand,  if  M  or  the  metacentre  is  below 
the  point  G,  as  at  Al',  for  instance,  the  buoyancy  of  the  fluid, 
which  will  then  be  exerted  according  to  M'Z',  will  cause  the 
straight  line  GB  to  depart  further  from  a  vertical  position,  and 
tend  to  upset  the  floating  body.  Therefore,  when  the  metacen- 
tre is  below  the  centre  of  gravity,  the  equilibrium  is  unstable  j 
and,  on  the  other  hand,  when  the  metacentre  is  above  the  centre 
of  gravity,  the  equilibrium  is  stable  ;  at  least  with  respect  to  all 
positions  in  which  the  plane  HFI  continues  vertical.  If,  in  a 
particular  case,  the  metacentre  coincides  with  the  centre  of  grav- 
ity, there  will  be  no  tendency  to  turn  one  way  or  the  other,  and 
the  straight  line  GB  will  remain  stationary,  whatever  inclination  be 
given  to  it. 


Equilibrium  of  Floating  Bodies.  327 

444.  When  the  form  of  the  floating  body  is  known,"  on  the 
supposition  that  its  position  is  very  near  that  of  equilibrium,  it 
will  be  easy  to  determine  the  place  of  the  metacentre,  or  rather 
to  determine  whether  this  point  is  above  or  below  the  centre  of 
gravity  of  the  body.  Let  us  suppose,  for  instance,  that  this  body 
is  a  homogeneous  horizontal  cylinder  of  an  elliptical  base,  and  of 
half   the  density   of  the   fluid  ;  let  HFM    be   a    vertical    section  Fig.2i5, 

.  .  .  216. 

made  at  equal  distances  from  the  two  bases  ;  in  the  position  of 
equilibrium,  one  of  the  two  axes  will  be  vertical ;  and  as  half 
of  the  bulk  will  be  immersed  in  the  fluid,  it  follows  that  the 
other  axis  will  coincide  with  the  water-line,  and  will  represent  the 
level  of  the  fluid.  The  vertical  axis  AF  is  the  transverse  in  figure 
215,  and  the  conjugate  in  figure  216.  Now  we  say  that  in  the 
first  case,  the  metacentre  is  below  the  centre  G  of  the  ellipse, 
which  is  also  the  centre  of  gravity  of  the  cylinder,  and  that  it  is 
above  it  in  the  second  case. 

Draw  through  the  point  G,  a  straight  line  A'F'  making  a  very 
small  angle  with  AF;  let  us  now  suppose  that  the  axis  AF  is. 
inclined  till  A^F'  becomes  vertical,  and  that  at  the  same  time  we 
raise  or  depress  a  very  little  the  centre  of  gravity  G,  so  that  the 
level  of  the  fluid  shall  become  the  straight  line  H^T,  perpendic- 
ular to  the  straight  line  A'F'  at  the  point  G\  In  this  position, 
the  part  H'FI'  of  the  ellipse  HFIA  will  be  immersed  in  the 
fluid;  and  this  part  is  divided  into  two  unequal  portions  H'F'G' 
and  TF'G^  by  the  straight  line  G'F'.  Now  it  is  evident  that  the 
centre  of  buoyancy  will  be  found  in  some  point  B',  situated  in  the 
greater  of  these  two  portions,  whence  it  is  evident  by  looking  at  the 
two  figures,  that  B'M,  parallel  to  the  straight  line  F'A',  will  meet 
the  straight  line  FA  at  the  point  M,  below  the  centre  of  gravity  in 
the  first  figure,  and  above  it  in  the  second. 

Thus  the  cylinder  which  we  are  considering  is  in  a  stable  or 
unstable  position  of  equilibrium,  according  as  the  conjugate  or 
transverse  axis  of  its  base  is  vertical.  Supposing  the  body  to  turn 
about  the  horizontal  straight  line  which  joins  the  centres  of  the  two 
bases,  it  will  pass  successively  through  four  positions  of  equilibrium, 
which  will  be  alternately  stable  and  unstable,  agreeably  to  the 
general  position  already  advanced. 


427. 


328  Hydrostatics. 

445.  To  fix,  in  a  few  simple  cases,  the  dimensions  of  the 
solid  and  its  relative  density  to  that  of  the  fluid  required  for  a 
particular  state  of  equilibrium,  let  the  body  in  question  be  a 
homogeneous  parallelopiped,  placed  vertically  in  the  fluid  ;  and  let 
Fig.217. 2)jPjg;  be  a  section  of  this  body  through  the  axis  parallel  to  one 
of  its  faces.  The  solid  will  evidently  sink  till  the  immersed  part 
JVF  shall  be  to  the  whole  height  AF,  as  its  density  is  to  the 
density  of  the  fluid  ;  and  its  centre  of  gravity  and  that  of  buoy- 
ancy will  be  G  and  B,  the  middle  |)oints  respectively  of  the  axis 
and  of  the  depressed  portion  A^F,  Suppose  now  that  the  body  is 
inclined  a  little,  shifting  its  water-line  from  the  position  HJVl  to 
H'Nl',  the  centre  of  buoyancy,  changing  from  B  to  B',  will  de- 
scribe a  small  arc  of  a  circle,  which  for  the  extent  under  considera- 
tion may  be  regarded  as  a  straight  line,  and  B'  will  be  raised  by 
a  quantity  which  will  be  to  the  altitude  PO  of  the  centre  of  grav- 
ity of  the  triangle  /A*Z',  as  the  area  of  the  rectangle  JV/F  is  to 
that  of  the  triangle  7AT,  that  is,  as  JVjP  :  \  II'.  Moreover  the 
horizontal  motion  of  B  will  be  to  J^P  or  |  JV7  in  the  same  pro- 
portion.    Whence 

NI  y  IT 
NF  :  Ijr  ::^m:BB'=  ^f^- 

But,  by  similar  triangles, 

3NF 


BB'  or  ^^l^r^  :BM  ::  II  :  JVI; 


accordingly  we  have,  for  the  height  of  the  metacentre   above  the 
centre  of  buoyancy, 

RM-    ^'   -     ^^ 
~  3  iVF  ""  12  NF' 

Let  AF,  the  height  of  the  parallelopiped,  be  denoted  by  h, 
its  breadth  or  thickness  HI  by  a,  and  its  density  or  specific 
gravity  by  A.  When  the  metacentre  coincides  with  the  centre 
of  gravity,  and  the   solid  floats  indifferently  in  any  position,  BM 

j^p ]^p 

IS  equal  to  BG  or  to ^ ;  that  is,  (water  being  the  fluid 

in  question,)  since  1,  the  density  of  the  fluid,  is  to  A,  the  density 
of  the  solid,  as  AF  or  R  is  to  J^F  or  a  h,  we  shall  have 


Equilibrium  of  Floating  Bodies.  329 

12  a  h  ~       2       '' 

2  a2  =  12  A  ^2  —  12  A^  h^ 
or 

A    —  A=:— g^. 

This  being  resolved  after  the  manner  of  an  equation  of  the  second 
degree,  gives 


=.*ji 


3  7^2  _  2  «2 


13  7*2      • 

If  the  parallelepiped  become  a  cube,   then  a  =z  h,   and  we 
have 


^  =  i  ±  Jr2  =  ^  ±  Jl  =  *  ^  ^=0'^=^»'^ 


29  near- 


ly ;  that  is,'  the  two  densities  are  0,79  and  0,21  nearly.  Between 
these  limits  there  can  be  no  stability  ;  but  above  0,79  or  below  0,21 
the  equilibrium  becomes  more  and  more  permanent.  Hence  a  cube 
of  beach  will  float  erect  in  water,  while  one  of  fir  or  cork  will  over- 
set ;  yet  all  these  three  cubes  will  stand  firmly  when  placed  upon 
the  surface  of  mercury.  We  restrict  ourselves,  in  this  illustration, 
to  cubes,  because  we  cannot  apply  the  remark  to  parallelepipeds 
generally.  A  stable  equilibrium  depends,  as  will  be  inferred  from 
what  has  been  said,  not  only  upon  the  relative  densities  of  the 
solid  and  fluid,  but  also  upon  the  proportion  between  the  hor- 
izontal and  vertical  dimensions  of  the  solid.  In  order  to  ascer- 
tain this  proportion  in  the  case  of  parallelepipeds,  and  on  the 
supposition  of  a  density  equal  to  half  that  of  the  fluid,  we  have  only 
to  put  equal  to  zero  the  radical  part  of  the  above  general  formula, 
and  we  shall  have 

2h^  =  2a^', 


accordingly 


7  O  I 

5  ==  I  =  if >  and  -  =  4  nearly. 


Whence,  approximatively, 
Mech.  42 


330  Hydrostatics. 

12  A  =  10  a, 

or 

A  :  a  : :  10  :  12, 

that  is,  a  parallelopiped  of  half  the  density  of  the  fluid,  and  having 
its  height  to  one  side  of  a  square  base,  as  10  to  12,  would  float 
indifibrently. 

44G.  But  if  the  relative  density  of  the  parallelopiped  were 
either  greater  or  less  than  -i,  its  equilibrium  would  beconne  stable. 
Thus,  if  we  suppose  a  equal  to  i,  we  shall  have  the  distance 
of  the  metacentre  above  the  centre  of  buoyancy,  as  follows, 
namely, 

^^=iijh=  12  xTx  10  "=  -  ^  ^'^  ^"^^^'^' 

and  for  the  distance  of  the  centre  of  gravity  above  the  centre  of 
buoyancy, 

BG  =  ^  ~-^  ^  =  1  A  =  1  10  =  3,3  inches ; 

so  that  the  centre  of  gravity  is  about  0,3  of  an  inch  below  the  meta- 
centre.    Therefore  the  equilibrium  would  be  stable. 

In  like  manner,  if  we  substitute  |  instead  of  ^  in  the  above 
equations,  or,  which  is  the  same  thing,  take  half  of  each  of  the 
above  results,  we  shall  have  |  of  0,3,  or  0,15  of  an  inch,  for  the 
distance  of  the  centre  of  gravity  below  the  metacentre. 

447.  These  principles  are  well  illustrated  by  the  masses  of 
ice  which  appear  on  the  rivers  of  the  colder  climates  at  the 
opening  of  spring.  Being  ordinarily  much  broader  than  they 
are  thick,  they  have  a  stable  equilibrium  in  Uieir  natural  position 
with  their  broad  surface  horizontal.  But  when  by  striking 
against  each  other,  or  by  passing  over  a  fall,  they  are  thrown  up 
sidewise,  their  equilibrium  becomes  unstable,  and  they  soon  re- 
turn to  their  former  position.  Moreover  a  piece  of  ice  of  a 
cubical  form  will  still  preserve  its  balance,  since  its  specific  grav- 
ity does  not  come  within  the  limits  already  pointed  out,  of  an  unsta- 
ble equilibrium.* 

*  The  specific  gravity  of  ice  is  0,92,  or,  compared  with  sea-water 
as  unity,  0,89. 


Equilibrium  of  Floating  Bodies.  331 

The  ice-bergs  that  float  from  the  polar  seas  down  into  warm- 
er regions,  are  gradually  dissolved,  not  only  by  the  sun's  rays,  but 
also  by  the  currents  of  warm  air  and  warm  water  to  which  they  are 
exposed.  But  these  causes  operate  more  powerfully  on  the  sides 
than  upon  the  top  and  bottom,  and  their  horizontal  dimensions  are 
thus  reduced  faster  than  their  vertical,  whereby  they  become  unsta- 
ble, and  are  overturned.  Being  still  subject  to  the  same  kind  of 
influence,  they  are  liable  to  repeated  and  frequent  changes  of  posi- 
tion before  they  are  completely  wasted. 

448.  To  investigate  generally  the  conditions  of  equilibrium  of 
a  floating  body,  let  JH^/F  represent  a  vertical  section,  the  point  (r  Fig.  218. 
of  the  principal  axis  ANF  being  the  centre  of  gravity  of  the 
whole,  and  the  point  B  the  centre  of  buoyancy.  The  solid  being 
inclined  a  litde,  the  water-line  HNI  shifts  to  H'NI',  and  the  centre 
of  buoyancy  B  to  B'.  From  what  has  been  said,  it  will  be  per- 
ceived that  the  area  of  HFI  is  to  the  sum  of  the  two  triangles 
NH,  HNH',  as  NP  is  to  BB' ;  that  is,  445. 

area  HFI  :  NI  X   IF  :  :  ^  NI  :  BB', 

but  the  triangles  INI,  BMB',  being  similar, 

n'  :  NL::  BB'  :  BM, 

therefore,  by  multiplying  the  terms  in  order,  and  suppressing  the 
common  factor  in  each  of  the  ratios,  we  have 

area  HFI  :  NI  :  :  |  iV7  :  BM, 

or,  calling  the  area  HFI,  a,  and  the  length  of  the  water-line,  HNI,  a, 
a:aay::^a:BM  =  ^^. 

Accordingly,  the  equilibrium  will  be  stable  when  the  cube  of  the 
length  of  the  water-line  NI,  divided  by  12  times  the  area  of  the 
section,  exceeds  die  interval  BG  between  the  centre  of  gravity 
and  that  of  buoyancy.  If  this  quotient  be  just  equal  to  BG,  the 
equilibrium  will  be  that  of  indifference  ;  and  lastly,  when  this  quo- 
tient is  less  than  BG,  the  equilibrium  will  be  unstable,  and  the  body 
will  be  liable  upon  a  slight  inclination  to  overset. 


333  Hydrostatics, 

Whatever  be  the  figure  of  the  section  HFI,  its  area  and  con- 
sequently the  centre  of  gravity  and  that  of  buoyancy  may  be 
found  to  any  required  degree  of  exactness  by  the  method  of 
article  114. 

Although  the  above  formula  has  reference  only  to  a  single 
lamina,  and  to  motion  in  the  plane  of  this  lamina,  it  is  still  ap- 
plicable to  any  solid  whose  parallel  sections  are  equal  and  sim- 
ilar, for  in  this  case  the  whole  may  be  considered  with  respect 
to  motion  in  a  parallel  plane,  as  concentrated  in  die  middle  sec- 
tion or  lamina  represented  by  HFI;  and  with  respect  to  motion 
in  a  vertical  plane  perpendicular  to  the  lamina,  by  supposing  a 
corresponding  section,  and  putting  a'  equal  to  the  area  of  this 
section,  and  a'  equal  to  the  length  of  the  water-line,  we  shall 
have  the  same  formula  to  express  the  conditions  of  equilibrium  as 
before. 

When  the  sections  or  laminae  are  unequal,  we  find   the  height 

of  the  metacentre  of  each  lamina,  and  multiply  it  by  the  bulk  of 

this  lamina,  and  then  divide  the  sum  of  the   products,  or  moments 

of  the   several  laminae,  by  the  sum  of  the  laminae  for  the  height  of 

76.    the  common  metacentre. 

449.  In  the  case  of  a  merchant  ship,  it  will  furnish  a  tolerable 
approximation  to  take  the  section  near  the  prow  where  the  girth  is 
commonly  the  largest.  The  transverse  section  of  the  hull  of  a 
ship  is  not  materially  different  from  the  form  of  a  parabola. 
Therefore,  on  this  supposition,  the  height  of  the  metacentre  above 
Fig.  219.  the  centre  of  buoyancy,  or  BM,  is  equal  to  the  cube  of  HI,  the 
Cat.  94.  length  of  the  water-line,  divided  by  twelve  times  the  area  of  HFI. 
But  the  area  HFI  is  equal  to  |  if/  X  iVjP.     Hence, 


12  ff  ~  12  X  I  ///  X  iVi^  ~~  8  iVi^  ~~  2  NF' 


*  Where  great  accuracy  is  required,  the  following  formula  may 
be  used ;  namely, 


Equilibrium  of  Floating  Bodies.  333 

that  is,  BM  is  equal  to  half  the  parameter  of  the  parabola.     B    Trig, 
being  the   centre   of   gravity  of  the    parabola    HFI,  its  height  is 
readily  found  to  be  |  NF.     Therefore  for  tlie  whole  height  of  the    107. 
raetacentre  above  the  keel,  we  have 

,  ^^   ,      ivT         6  NF"-^  5  NI 
3   l\Tr  J -—  ! — , 

'  ^2  NF  10  NF 

Such  is  the  height  of  the  metacentre  above  the  keel,  on  the 
supposition  that  the  vertical  sections  are  all  equal  and  parabolic, 
which  is  nearly  the  case  with  respect  to  long  track-boats.  But 
the  figure  of  the  keel  in  most  vessels,  fitted  for  sailing,  approach- 
es to  a  serai-ellipse,  which  is  likewise  the  general  form  of  a  hori- 
zontal section.  Owing  to  these  modifications,  the  metacentre  is 
found  to  be  depressed  about  one  fourth  part,  and  consequently  its 
height  above  the  centre  of  buoyancy  will  be 

2  — 2 

3  y      HI    _3NI 

^  ^   8  NF  ~  S  NF' 

In  a  ship,  for  example,  whose  water-line  is  40  feet,  and  the  depth  of 
its  immersed  portion  15  feet,  we  shall  have  for  the  height  of  the 
metacentre  above  the  centre  of  buoyancy, 

^_J^^  _  ij!_o_o  —  10  feet 
8  X  15  ~    '^»    ~ 

But  the  centre  of  gravity  of  the  immersed  part  is  |  1 5  or  6  feet 
below  the  water-line.  Hence  the  metacentre  is  4  feet  above 
the  water-line.  The  ship  will  therefore  float  securely  so  long 
as   the    general   centre  of   gravity  is  kept  under  that  limit.      In 


in  which  T  iY/ represents  the  sum  of  the  cubes  of  the  perpendicu- 
lars RQ,  ON,  &/C.,  of  figure  50,  these  perpendiculars  being  taken 
at  equal  distances  and  so  near  to  each  other  that  the  included 
portions  of  the  curve  ON,  OK,  &lc.,  may  be  considered  as  straight 
lines,  the  common  distance  being  denoted  by  c,  and  the  bulk  of 
the  immersed  part  of  the  vessel  by  6.  The  investigation  of  this 
formula  is  very  simple,  and  is  omitted  here  merely  on  account  of 
its  length.     See  Bczout's  Mecaniquc,  art.  359. 


334  Hydrostatics. 

loaded  vessels,  the  centre  of  gravity  has  commonly  been  found  to 
be  higher  than  the  centre  of  buoyancy,  by  about  the  eighth  part 
of  the  extreme  breadth.  Accordingly,  in  the  present  instance, 
the  centre  of  gravity  of  the  whole  mass  would  still  be  one 
foot  below  the  surface  of  the  water,  or  five  feet  lower  than  the 
metacentre,  which  would  be  amply  sufficient  for  the  stability  of  the 
ship. 

450.  Such  is  the  position  of  the  metacentre  in  the  vertical 
plane  at  right  angles  to  the  longitudinal  axis,  and  which  reg- 
ulates the  rolling  of  a  vessel  from  side  to  side.  But  there  is 
anodier  similar  point  in  the  plane  of  the  masts  and  keel,  which 
determines  the  pitching,  or  the  movement  of  alternate  rising  and 
sinking  of  the  prow.  The  height  of  this  metacentre  is  derived 
from  the  same  formuka,  by  substituting  only  the  length,  for  the 
breadth  of  the  vessel.  Thus,  let  the  keel  measure  ISO  feet,  and 
w'e  have 

?|«a!  =  202i  feet. 

With  such  a  strong  tendency  to  stability,  therefore,  in  the  direction 
of  its  course,  a  ship  can  scarcely  ever  founder  in  consequence  of 
pitching  at  sea. 

The  formula  now  given  for  computing  the  height  of  the  me- 
tacentre above  the  centre  of  buoyancy,  may,  with  some  modifi- 
cation, be  deemed  sufficiently  accurate  in  practice.  It  is  best 
adapted,  however,  for  cutters  or  frigates,  and  will  require  to  be 
somewhat  diminished  in  the  case  of  merchant  vessels.  Mr.  At- 
wood  performed  a  laborious  calculation  on  the  hull  of  the  CufF- 
nells,  a  ship  built  for  the  service  of  the  East  India  Company, 
having  divided  it  into  34  transverse  sections,  of  five  feet  interval. 
The  result  was,  that  the  metacentre  stood  only  4  feet  3  inches 
above  the  centre  of  buoyancy.  But  that  ship,  being  designed 
chiefly  for  burthen,  appears  from  the  drawings  to  have  been 
constructed  after  a  very  heavy  model,  its  vertical  sections  ap- 
proaching much  nearer  to  rectangles  than  parabolas.  To  suit  it, 
the  formula  above  given  would  have  required  to  be  reduced  two 

o 

NI 

thirds,  or  to  .     Now  the  breadth  of  the  principal  section  was 


Equilibrium  of  Floating  Bodies.  335 

43  feet  and  two  inches,  and  its  depth  22  feet  9  inches.     Whence 

(21  6)2 

■    q\     ■  =5,1  feet,  differing  little  from  the  conclusion  of  a  stricter 

but  very  tedious  process. 

451.  Since  the  height  of  the  metacentre  is  inversely  as  the 
draught  of  a  vessel,  and  directly  as  the  square  of  its  breadth,  its 
stability  depends  mainly  on  its  spreading  shape.  This  property 
is  an  essential  condition  in  the  construction  of  life-boats.  But 
the  lowering  even  of  the  centre  of  gravity  has  been  found  to  be 
sometimes  insufficient  to  procure  stability  to  new  ships,  which, 
after  various  ineffectual  attempts,  were  rendered  serviceable,  by 
applying  a  sheathing  of  light  wood  along  the  outside,  and  thus 
widening  the  plane  of  floating. 

452.  It  is  not  very  difficult  to  determine  the  centre  of  buoy- 
ancy, by  gauging  the  immersed  part  of  the  hull.  A  cubic  foot 
of  sea-water  weighs  64"'-  avoirdupois,  and  35  feet,  therefore,  make 
a  ton.  The  load  of  the  vessel  corresponding  to  every  draught  of 
water  may  be  hence  computed. 

453.  The  height  of  the  metacentre  above  the  centre  of  grav-  Fig.  220. 
ity  in  a  loaded  vessel,  may  by  determined   by  simple  observation. 

Let  a  long,  stiff,  and  light  beam  be  projected  transversely  from 
the  middle  of  the  deck,  and  a  heavy  weight  suspended  from  its 
remote  end,  inclining  the  ship  to  a  certain  angle,  which  is  easily 
measured.  Thus,  if  JVL  represent  this  lever,  q  the  weight  at- 
tached, M  the  metacentre,  and  GM^  the  inclination  produced, 
G  being  the  centre  of  gravity,  and  GR  a  perpendicular  drawn  from 
it  to  the  vertical  L  q,  the  power  of  the  weight  q  to  incline  the 
vessel  will  be  expressed  by  q  X  GR  ;  but  p  denoting  the  entire 
weight  of  die  vessel,  the  effort  exerted  at  the  metacentre  to  keep 
the  mast  erect,  will  be  represented  by 

p  X  GQ,  or  p  X  GM  X  sin  GMQ^. 

Wherefore 

q  X  GR=p  X  GM  X  sin  GMq, 


and  consequently  the  elevation  GM  above   the  centre  of  gravity 
is  expressed  by   -  .  -. — qWq'     ^°^  ^^  "^^^'  ^•^''^'^"^  ^"X  ^^"" 


336  Hydrostatics. 

sible  error,  be  assumed  as  equal  to  the  length  LJY  of  the  beam 
from  the  middle  of  the  deck.  Supposing  the  height  of  the  me- 
tacentre  to  be  3  feet  lOi  inches  above  the  centre  of  gravity,  a 
weight  equal  to  the  two  hundredth  part  of  the  burthen  or  ton- 
nage of  the  ship,  and  acting  on  a  lever  of  50  feet  in  length,  would 
occasion  an  inclination  of  five  degrees.  If  the  experiment  were 
performed  in  a  wet-dock,  or  on  a  smooth  calm  sea,  such  a  small 
angle  could  be  measured  with  sufficient  accuracy.  In  calculating 
the  effect  of  this  disturbing  influence,  it  is  easy  to  perceive  that 
half  the  weight  of  the  beam  should  be  added  to  q.  A  trifling 
correction  may  be  likewise  made,  for  assuming  GR  as  equal  to 
JVL ;  by  first  diminishing  JVL,  by  its  product  into  the  versed  sine 
of  the  inclination,  and  next  augmenting  it,  by  the  product  of  GJV 
into  the  sine  of  that  angle. 

A  similar  method  might  be  adopted  to  discover  the  height 
of  the  longitudinal  metacentre  of  the  ship,  above  the  common 
centre  of  gravity.  But,  acting  in  this  direction,  a  greater  load 
will  be  required  to  produce  a  sensible  depression.  Let  such  a 
load  be  carried  to  the  prow  of  the  vessel,  and  again  transferred 
to  the  stern.  The  intermediate  place  of  the  centre  of  gravity 
is  hence  determined,  for  its  distances  from  these  opposite  points 
of  pressure  must  evidently  be  inversely  as  the  corresponding  an- 
gles of  inclination.  The  small  change  of  the  centre  of  gravity 
occasioned  by  the  interchange  of  these  loads,  may  likewise  be 
computed.  Finally,  therefore,  the  product  of  either  load  into  its 
distance  from  the  centre  of  gravity,  being  divided  by  the  product 
of  the  whole  burthen  of  the  ship  into  the  sine  of  the  inclination, 
will  give  the  height  of  the  metacentre  of  the  longitudinal  section  on 
which  depends  the  motion  of  pitching. 


Capillary  Attraction. 


454.  The  most  curious  natural   phenomena  are  those  which 
make    us    acquainted    with   the   intimate    constitution    of   bodies 


Capillary  Attraction.  gS^ 

and  the  reciprocal  action  which  their  particles  exert  upon 
each  other.  We  come  now  to  consider  a  class  of  these  phenom- 
ena of  considerable  extent  and  variety,  and  which  are  the  more 
deserving  of  attention,  as  they  are  susceptible  of  a  rigorous 
calculation. 

If  a  disk  of  glass,  marble,  or  metal,  &c.,  be  suspended  to  the 
scale  of  a  balance,  and  counterpoised  by  an  equal  weight  in  the 
opposite  scale,  upon  being  made  to  touch  the  surface  of  a  liquid 
capable  of  moistening  it,  it  will  be  found  to  adhere  with  a  certain 
force,  and  to  require  an  additional  weight  in  the  opposite  scale  of 
the  balance  to  detach  it.  This  adhesion  is  not  produced  by  the 
pressure  of  the  air,  for  it  takes  place  equally  well  in  a  vacuum. 
We  infer,  therefore,  that  it  is  the  particles  of  the  solid  which  attach 
themselves  to  the  particles  of  the  fluid  by  virtue  of  a  force  of  af- 
finiiy.  But  there  is  to  be  inferred  also  a  similar  action  between 
the  particles  of  the  fluid  itself.  Indeed  when  the  disk  is  capable 
of  being  moistened  by  the  liquid,  as  is  the  case  when  glass  is 
used  with  water  or  alcohol,  the  disk,  upon  being  withdrawn 
brings  with  ij;  a  small  liquid  film,  or  lamina,  which  adheres  to  it. 
It  is  not  then,  strictly  speaking,  the  solid  which  is  detached  from 
the  liquid,  it  is  this  small  lamina  which  is  separated  from 
the  particles  immediately  below  it.  Now  the  force  employed 
thus  to  detach  il,  is  incomparably  more  considerable  than  the 
proper  weight  of  this  lamina ;  consequently  the  excess  of  force 
proves  the  existence  of  an  internal  adhesion  in  the  liquid  which 
would  keep  the  small  lamina  united  to  the  rest  of  the  liquid  mass 
independently  of  gravity. 

According  to  the  notions  which  we  have  formed  of  the  re- 
ciprocal action  of  the  particles  of  bodies  upon  each  other, 
the  force  in  question  seems  to  be  of  the  same  nature  and  to 
have  a  sensible  effect  only  at  very  small  distances.  This  is 
moreover  proved  by  experiment.  Whatever  be  the  thickness 
of  the  disk,  so  long  as  the  form  and  substance  are  the  same,  the 
force  required  to  detach  it  from  a  given  liquid,  is  also  the  same. 
Accordingly,  beyond  a  certain  thickness,  probably  less  than  any 
within  the  reach  of  human  art  to  attain,  any  augmentation  has 
no  effect  capable  of  being  appreciated.  Whence  it  will  be  seen 
that  this  action  is  not  capable  of  producing  sensible  effects,  ex- 
Mech.  43 


338  Hydrostatics. 

cept  at  distances  extremely  small.  But  as  a  further  proof,  it 
may  be  mentioned  that  all  disks  of  the  same  size,  whatever  be 
the  substance,  provided  it  is  capable  of  being  moistened  by  the 
liquid,  require  precisely  the  same  force  to  detach  them ;  so  that 
in  these  cases  the  thin  film  of  water,  which  attaches  itself  to  their 
surfaces,  places  these  surfaces  and  the  rest  of  the  fluid  at  inter- 
vals sufficiently  great  to  prevent  any  sensible  action  taking  place  ; 
and  the  force  required  to  detach  all  disks  of  the  same  size,  what- 
ever be  the  substance,  is  precisely  the  same,  since  it  is  that 
which  is  necessary  to  detach  the  liquid  from  itself. 

455.  Phenomena  arising  from  the  same  cause,  but  differing 
Fig.22l.in  appearance,  are  also  observed  when  tubes  of  a  small  bore 
are  immersed  in  a  liquid.  If  the  liquid  is  of  a  nature  to  moisten 
the  tube,  it  will  be  found  to  ascend  into  the  interior,  and  to  main- 
tain itself  above  the  natural  level.  Wlien  glass  tubes,  for  exam- 
ple, of  a  fine  bore  are  immersed  endwise  in  water  or  alcohol,  this 
elevation  of  the  fluid  will  take  place ;  and  in  these  cases,  the 
upper  extremity  of  the  column  is  concave.  But  if  the  liquid  is 
not  of  a  nature  to  moisten  the  tube,  as  is  the  case  with  mercury, 
melted  lead,  kc,  used  with  glass  taken  in  its  ordinary  state,  the 
liquid  in  the  tube  will  be  depressed  instead  of  being  elevated, 
and  the  upper  extremity  of  the  column  will  be  convex.  In  all 
these  cases  the  elevation  or  depression  is  the  more  considerable 
according  as  the  diameter  of  the  bore  is  less.  Such  are  the 
phenomena  which  are  called  capillary  from  the  circumstance  of 
the  fineness  of  the  bore  of  the  tube. 

The  phenomena  being  the  same  in  a  vacuum  as  in  the  open 
air,  they  are  not  connected  with  the  pressure  of  the  atmosphere.  * 

*  In  the  discussion  formerly  maintained  upon  this  subject,  a 
perplexing  fact  was  stated  ;  namely,  that  if  a  glass  tube  consisting 
of  two  cylinders  of  different  bores,  joined  endwise,  be  immersed 
in  water,  the  larger  end  being  downward,  so  as  to  cause  the  fluid 
to  rise  into  the  smaller  part  of  the  tube,  the  column  sustained  will 
be  of  the  same  length  as  in  a  tube  whose  bore  is  throughout  of  the 
same  size  with  this  smaller.  The  experiment  is  now  found  not  to 
succeed  in  a  vacuum.  The  peculiarity  of  the  phenomenon,  there- 
fore, must  depend  upon  the  pressure  of  the  atmosphere. 


Capillary  Attraction.  339 

But  they  depend,  like  the  preceding,  upon  the  attraction  exerted 
by  the  tube  upon  the  liquid  and  by  the  liquid  upon  itself;  so  that 
when  the  thickness  of  the  tube  is  made  to  vary,  the  bore  remaining 
unchanged,  the  elevations  and  depressions  of  the  liquid  remain 
the  same,  which  proves  that  beyond  a  certain  thickness,  proba- 
bly too  small  for  us  to  attain,  any  additional  matter  that  may 
be  accumulated  will  have  no  appreciable  effect.  It  follows  from 
this  law,  that  when  tubes  of  the  same  diameter  are  com- 
pletely moistened  throughout  by  the  liquid,  the  elevation  or  depres- 
sion will  be  the  same  in  all,  whatever  the  substance  of  the 
tube,  which  shows  that  the  thin  film  attached  to  the  interior 
surface,  removes  by  its  interposition  the  rest  of  the  liquid  mass 
so  as  to  render  the  attraction  of  the  tube  insensible  ;  consequendy 
the  elevation  is  the  same  in  all  tubes  of  the  same  bore, 
because  it  is  equal  to  that  which  would  proceed  from  a  tube  of  the 
same  diameter  formed  of  the  liquid  itself.* 

456.  Setting  out  from  the  results  furnished  by  the  calculus, 
we  are  able  to  give  a  satisfactory  explanation  of  the  phenomena 
of  capillary  tubes.  Beginning  with  the  case  in  which  the  fluid  is 
elevated  above  the  natural  level,  and  which  requires  the  upper 
extremity  of  the  fluid  colunm  to  be  concave,  we  suppose  an 
infinitely  small  filament  of  fluid  extending  from  the  lowest  point  Fig.222. 
of  the  meniscus  along  the  axis  of  the  tube,  and  then  returning 
in  any  manner  through  the  mass  of  the  liquid  to  the  free  surface. 
The  fluid  being  in  a   stale  of  equilibrium,  this  filament  will  be  in 

*  The  diameter  of  the  bore  of  a  tube  is  found  by  first  weighing 
the  tube  empty,  and  then  after  having  introduced  a  certain  quantity 
of  mercury,  weighing  it  again.  The  excess  of  the  latter  weight 
above  the  former  will  be  the  weight  of  the  column  of  mercury.  By 
calling  this  weight  w,  the  length  of  the  column  /,  and  the  radius  of 
the  bore  R,  n  being  the  ratio  of  the  circumference  of  a  circle  to  itsfieoin. 
diameter,  we  shall  have  for  the  bulk  of  mercury  contained  in  the 
tube  n  V?  I.  If  w  be  the  weight  of  a  cubic  inch  of  mercury  at  the 
temperature  assumed  in  the  experiment,  and  r  and  I  be  also  ex- 
pressed in  inches,  or  parts  of  an  inch,  10'  n -e?  I  will  be  the  weight 
of  the  column   in  question;  whence 


w' 


nv?lz=i  IV,    and  r  :=     \— — 7. 


340  Hydrostatics. 

a  state  of  equilibrium.  But  it  is  pressed  downward  at  the  two 
extremities  with  unequal  forces.  The  force  exerted  at  the  free 
surface  is  the  action  of  a  body  terminated  by  a  plane  surface  ; 
the  other  in  the  interior  of  the  tube  is  the  action  of  the  same 
body  terminated  by  a  concave  surface,  or  one  in  which  there , 
is  a  contrary  attraction  upward,  the  little  annulus  cut  off  by  a 
horizontal  plane  passing  through  the  lowest  point  of  the  meniscus, 
and  which  is  supported  by  the  attraction  of  the  glass,  exerting 
an  upward  force.  It  is  necessary,  therefore,  in  order  that  an 
equilibrium  may  take  place  that  the  fluid  should  rise  in  the  tube 
till  the  weight  of  the  column  thus  elevated  above  the  natural 
level  should  compensate  for  this  difference  in  the  downward 
pressures  exerted  at  the  two  extremities  of  the  filament.  This 
difference  is  in  the  inverse  ratio  of  the  diameter  of  the  tube  ;  the 
height  of  the  small  column  must  accordingly  be  in  the  same  ratio  ; 
and  this  is  conformable  to  the  results  of  our  observation. 

457.  The  heights  to  which  water  and  alcohol  ascend  in  ca- 
pillary tubes  were  observed  by  M.  Gay  Lussac  with  the  greatest 
care.     The  following  are  a  few  of  his  results. 

Water. 

Diameter  of  the  tube.  Height  to  the  lowest  point  Temperature, 

of  the  concavity. 

(1.)  1,29441*  23,1634  47,5°  Fah. 

(2.)  l,903Sl  15,5861  47,5°. 

Alcohol,  (specific  gravity  being  0.81961.) 

Diameter  of  the  tube.  Height  to  the  lowest  point  Temperature, 

of  the  concavity. 

(1.)  1,29444  9,18235  47,5° 

(2.)  1,90381  6,08397  47,5°. 

M.  Ga}^  Lussac  measured  also  the  ascent  of  water  between  two 
plates  of  glass   ground    perfectly   plane,   and    placed    exactly  par- 


*  The  measures  of  M.  Gay  Lussac  are  given  in  millimetres  or 
0,039371  of  an  inch.  The  results  in  which  we  are  principally 
concerned,  are  reduced  to  English  inches. 


Capillary  Attraction.  341 

allel  to  each   other.     The  result  of  his  observations  was  as  fol-Fig.224. 
lows. 

Distance  of  the  plates.  Height  to  the  lowest  point  Temperature. 

of  the  concavity. 

1,0G9  13,574  62°. 

458.  Let  AB  he  a  vertical  lube  whose  sides  are  perpendicular  Fig.222. 
to  its  base,  and  which  is  immersed  in  a  fluid  tiiat  rises  in  the  inte- 
rior of  the  tube  above  its  natural  level.  A  thin  film  of  fluid  is 
first  raised  by  the  action  of  the  sides  of  the  tube  ;  this  film  raises 
a  second  film,  and  this  second  a  third,  till  the  weight  of  the  vol- 
ume of  fluid  raised  exactly  balances  all  the  forces  by  which  it 
is  actuated.  Hence  it  is  obvious,  that  the  elevation  of  the  col- 
umn is  produced  by  the  attraction  of  the  tube  for  the  fluid, 
and  the  attraction  of  the  fluid  for  itself.  Let  us  suppose  that  the 
inner  surface  of  the  tube  AB  is  prolonged  to  E,  and  after  bend- 
ing itself  horizontally  in  the  direction  ED,  that  it  assumes  a 
vertical  direction  DC;  and  let  us  suppose  the  sides  of  this  tube 
to  be  formed  of  a  film  of  ice,  or  to  be  so  extremely  thin,  as  not 
to  have  any  action  on  the  fluid  which  it  contains,  and  not  to  pre- 
vent the  reciprocal  action  which  takes  place  between  the  parti- 
cles of  the  first  tube  JIB  and  the  particles  of  the  fluid.  Now, 
since  the  fluid  in  the  tubes  AE,  CD,  is  in  equilibrium,  it  is  ob- 
vious, that  the  excess  of  pressure  of  the  fluid  in  AE  is  destroy- 
ed by  the  vertical  attraction  of  the  tube  and  of  the  fluid  upon 
the  fluid  contained  in  AB.  In  analysing  these  different  attrac- 
tions, Laplace  considers  first  those  which  take  place  under  the 
tube  AB.  The  fluid  column  BE  is  attracted,  (1.)  by  itself; 
(2.)  by  the  fluid  surrounding  the  tube  BE.  But  these  two  attrac- 
tions are  destroyed  by  the  similar  attraction  experienced  by  the  fluid 
contained  in  the  branch  DC,  so  that  they  may  be  entirely  neg- 
lected. The  fluid  in  BE  is  also  attracted  vertically  upward  by 
the  fluid  in  AB ;  but  this  attraction  is  destroyed  by  the  attraction 
which  the  fluid  in  BE  exerts  in  turn  upon  that  in  AB,  so  that 
these  balanced  attractions  may  likewise  be  neglected.  The  fluid 
in  BE  is  likewise  attracted  vertically  upwards  by  the  tube  AB, 
with  a  force  which  we  shall  call  q,  and  which  contributes  to  destroy 
the  excess  of  pressure  exerted  upon  it  by  the  column  BF,  raised 
in  the  tube  above  its  natural  level. 


342  Hydrostatics. 

Now  riie  fluid  in  ihe  lower  part  of  the  round  tube  AB  is  at- 
tracted, (1.)  by  itself;  but  the  reciprocal  attractions  of  a  body  do 
not  communicate  to  it  any  motion,  if  it  is  solid,  and  we  may,  with- 
out disturbing  the  equilibrium,  conceive  the  fluid  in  AB  frozen. 
(2.)  The  fluid  in  the  lower  part  of  AB  is  attracted  by  the  fluid 
within  the  tube  BE ;  but  as  the  fluid  of  the  tube  BE  is  attract- 
ed upwards  by  the  same  force,  these  two  actions  may  be  neg- 
lected as  balancing  each  other.  (3.)  The  fluid  in  the  lower  part 
of  AB  is  attracted  by  the  fluid  which  surrounds  the  ideal  tube 
BE,  and  the  result  of  this  attraction  is  a  vertical  force  acting 
downwards,  which  we  may  call  —  q',  the  contrary  sign  being 
applied,  as  the  force  is  here  opposite  to  the  other  force  q.  As 
it  is  highly  probable  that  the  attractive  forces  exerted  by  the 
glass  and  the  water  vary  according  to  the  same  function  of  the 
distance,  so  as  to  differ  only  in  their  magnitude,  we  may  employ 
the  constant  coefficients  p,  p',  as  measures  of  their  intensity,  so 
that  the  forces  g-,  - — q',  will  be  proportional  top,  p' ;  for  the  in- 
terior surface  of  the  fluid  which  surrounds  the  tube  BE,  is  the 
same  as  the  interior  surface  of  the  tube  AB.  Consequently,  the 
two  masses,  namely,  the  glass  in  AB,  and  the  fluid  around  BE, 
differ  only  in  their  thickness ;  but  as  the  attraction  of  both  these 
masses  is  insensible  at  sensible  distances,  the  difference  of  their 
thicknesses,  provided  their  thicknesses  are  sensible,  will  produce 
no  difference  in  the  attractions.  (4.)  The  fluid  in  the  tube  AB  is 
also  acted  upon  by  another  force,  namely,  by  the  sides  of  the 
tube  AB  in  which  it  is  enclosed.  If  we  conceive  the  column  FB 
divided  into  an  infinite  number  of  elementary  vertical  columns, 
and  if,  at  the  upper  extremity  of  one  of  these  columns,  we  draw  a 
horizontal  plane,  the  portion  of  the  tube  comprehended  between 
this  plane  and  the  level  surface  5 C  of  the  fluid,  will  not  produce 
any  vertical  force  upon  the  column  ;  consequenfly,  the  only  effec- 
tive vertical  force  is  that  which  is  produced  by  the  ring  of  the 
85,  tube  immediately  above  the  horizontal  plane.  Now  the  vertical 
attraction  of  this  part  of  the  tube  upon  BE,  will  bs  equal  to  that 
of  the  entire  tube  upon  the  column  BE,  which  is  equal  in  diam- 
eter, and  similarly  placed.  This  new  force  will  therefore  be 
represented  by  -\-  q.  In  combining  these  different  forces,  it  is 
manifest  that  the  fluid  column  BF  is  attracted  upwards  by  the 
two   forces    -\-  q^     +  ?»   and    downwards   by    the    force  —  q' ; 


Capillary  Attraction.  343 

consequently,  the  force  with  which  it  is  elevated  will  be  2  5*  —  q'. 
If  we  represent  the  bulk  of  the  column  BF  by  6,  its  density 
by  A,  and  the  force  of  gravity  by  g,  then  g  ^b  will  represent 
the  weight  of  the  elevated  column  ;  but,  as  this  weight  is  in 
equilibrium  with  the  forces  by  which  it  is  raised,  we  shall  have 
the  following  equation  ; 

g  Ah  ■=.  2  q  —  q'. 

If  the  force  2  17  is  less  than  5',  then  h  will  be  negative,  and  the 
fluid  will  be  depressed  in  the  tube  ;  but  as  long  as  2  g'  is  greater 
than  q',  h  will  be  positive,  and  the  fluid  will  rise  above  its  natural 
level. 

Since  the  attractive  forces,  both  of  the  glass  and  fluid,  are 
insensible  at  sensible  distances,  the  surface  of  the  tube  AB  will  act 
sensibly  only  on  the  film  of  fluid  immediately  in  contact  with 
it.  We  may  therefore  neglect  the  consideration  of  the  curvature, 
and  consider  the  inner  surface  as  developed  upon  a  plane.  The 
force  q  will  therefore  be  proportional  to  the  width  of  this  plane, 
or,  which  is  the  same  thing,  to  the  interior  circumference  of  the 
tube.  Calling  c,  therefore,  the  circumference  of  the  tube,  we 
shall  have  q  =  p  c,  p  being  a  constant  quantity  representing 
the  force  of  attraction  of  the  tube  AB  for  the  fluid,  in  the 
case  where  the  attractions  of  different  bodies  are  expressed 
by  the  same  function  of  the  distance.  In  every  case,  however, 
p  expresses  a  quantity  dependent  on  the  atlraction  of  the  matter, 
of  the  tube,  and  independent  of  its  figure  and  magnitude.  In 
like  manner  we  shall  have  q'  =z  p'  c  ',  p'  expressing  the  same 
thing  with  regard  to  the  attraction  of  the  fluid  for  itself,  thatj^ 
expresses  with  regard  to  the  attraction  of  the  tube  for  the  fluid. 
By  substituting  these  values  of  q,  q',  in  the  preceding  equation, 
we  shall  have 

g  A  b  =2  p  c  —  p'  c  =  c  {2  p  —  p')  (i.) 

If  we  now  substitute,  in  this  general  formula,  the  value  of  c  in 
terms  of  the  radius,  if  it  is  a  capillary  tube,  or  in  terms  of  the  sides, 
if  the  section  is  a  rectangle,  and  the  value  of  6  in  terms  of  the  radius 
and  altitude  of  the  fluid  column,  we  shall  obtain  an  equation  by 
which  the  heights  of  ascent  may  be  calculated  for  tubes  of  all 
diameters,  when  the  height,  belonging  to  any  given  diameter,  has 
been  ascertained  by  direct  experiment. 


555 


344  Hydrostatics. 

In  the  case  of  a  cylindrical  tube,  let  n  represent  the  ratio 
of  the  circumference  to  the  diameter,  h  the  height  of  the  fluid 
column  reckoned  from  the  lowest  point  of  the  meniscus,  h'  the 
mean  height  to  which  the  fluid  rises,  or  the  height  at  which  the 
fluid  would  stand  if  the  meniscus  were  to  settle  down  and  assume 
a  level  surface  ;  then  we  have  n  n^  for  the  solid  contents  of  a 
Geom.  cylinder  of  the  same  height  and  radius  as  the  meniscus ;  and  as 
the  meniscus,  added  to  the  solid  contents  of  a  hemisphere  of 
the  same  radius,  must  be  equal  to  n  k^,  (or  in  other  words,  the 
cylinder  n  r^,   diminished   by  the   hemisplicre    |  n  r^,   is  equal   to 

the  mensicus,)  we  have  n  v? 1^ —  ,  or  -^,  for  the  solid  con- 
tents of  the  meniscus.  But  since  — jr—  :=  ti  r^  X  ri?  it  follows 
that   the    meniscus   —^r-    is    equal   to    a    cylinder  whose   base    is 

71 R^,  and  altitude  ^.     Hence,  we  have  A'  =r  /t  -|-  -  ;  or,  which   is 

the  same  thing,  the  mean  altitude  h'  is  always  equal  to  the  altitude 
h  of  the  lower  point  of  the  concavity  of  the  meniscus,  increased 
by  one  third  of  the  radius,  or  one  sixth  of  the  diameter  of  the 
capillary  tube.  Now,  since  the  contour  c  of  the  tube  =  2  ti  r, 
and  since  the  bulk  h  of  water  raised  is  equal  to  A'  X  tt  r^,  we 
have,    by   substituting    these    values    in    the   general    formula    (i.) 

g  Ah^  n  R^  =  2  71  R  {2p  —  //)  (ii.) 

and,  dividing  by  ti  r  and  g  a,  we  obtain, 

/,.  K  =  2  2£-y    ^^^  ^,  ^  2  ^^^  X  ^         (III.) 
g  A     '  g  A  R  ^       ^ 

In  applying  this  formula  to  M.  Gay  Lussac's  experiments,  we  have 
2  ^P—P'  _  r/j/  _  0,647205  X  (23,1634  +  0,215735) 

=  15,131 1,  or  0,023454  of  an  inch  for  the  first  experiment;  and, 
since  the  heights  are  inversely  as  the  radii  or  diameters,  0,023454  or 
its  double  0,046908  is  a  constant  quantity.  In  order  to  f]nd  the 
height  of  the  fluid  in  the  second  tube  by  means  of  diis  constant 
quantity,  we  have 


Capillary  Attraction.  345 

,  _L90381.  ^  o  951905, 


and 


2  ^P-^^  ^^  ~-h'-  i^J^  -  15  8956 
^      gA       ^   K-^   -  0,951905  -  1^'^y^^' 

from  which  if  we  subtract  one  sixth  of  the  diameter,  or  0,3173, 
we  have  15,5783  for  the  altitude  h  of  the  lowest  point  of  the  con- 
cavity of  the  meniscus,  which  differs  only  0,0078  or  0,0003  of  an 
inch  from  15,5861,  the  observed  altitude. 

If  we  apply  the  same  formula  to  M.  Gay  Lussac's  experiments 
on  alcohol,  we  shall  find  for  the  constant  quantity 

2  ^J-=:-£  =  6,0825, 
g  A 

as  deduced  from  the  first  experiment,  and  A  =  6,0725,  which  dif- 
fers only  0,01147,  or  0,00045  of  an  inch  from  6,08397,  the  ob- 
served altitude. 

From  these  examples,  it  will  be  seen  that  the  mean  altitudes, 
or  the  values  of  h\  are  reciprocally  proportional  to  the  diameters 
of  the  lubes  very  nearly  ;  and  that  inaccurate  experiments,  the 
correction  made  by  the  addition  of  the  sixth  part  of  the  diame- 
ter of  the  tube  is  indispensably  requisite. 

459.  If  the  section  of  the  bore  in  which  the  fluid  ascends  is  a^ig-224. 
rectangle,  whose  greater  side  is  a,  and  smaller  side  d,  the  base 
of  the  elevated  column  will  be  a  S,  and  its  perimeter  2  a  -\-  2  S. 
Then  the  meniscus  will  be  equal  to  the  small  rectangular  prism, 
whose  base  is  a  d  and  height  i  d,  minus  the  semicylinder  whose 
radius  \s  ^  8  and  length  a ;  accordingly,  we  have  for  the  solidity  of 
the  meniscus. 


a  32         a  TT  52 

~2  8~ 


=  ^0-0- 


that  is. 


'=''  +  K'-^> 


Hence,  if  in  the  general  (i.)  equation  we  substitute  for  c  its  equal 
Mech.  44 


346  Hydrostatics. 

2  a  +  2  (5, 

and  for  b  its  equal  a  d  h',  we  shall  have 

g  Ah'ad  =  (2p—p')  X  (2  a  +  2  5) ; 

and,  dividing  by  a  and  by  g  A,  we  have 

g  A 

and 


2p 


A/  =  2  X   -^^ 


0+D 


In  applying  this  formula  to  the  elevation  of  water  between  two 
glass  plates,  the  side  a  is  very  great  compared  with  d,  and  there- 

fore  the  quantity  —  being  almost  insensible,  may  be  safely  neglected. 

Hence  the  formula  becomes 

^  A  <5 

By  comparing  this  formula  with  the  formula  (in.)  it  is  evident 
that  water  will  rise  to  the  same  height  between  glass  plates,  as  in 
a  tube,  provided  the  distance  8  between  the  two  plates  is  equal 
to  K,  or  half  the  diameter  of  the  tube  ;  in  other  words,  that,  when 
the  distance  between  the  plates  is  equal  to  the  diameter  .of  the 
tube,  the  elevation  in  the  former  case  is  half  that  in  the  latter. 
This  result  was  obtained  by  Newton,  and  has  been  confirmed  by 
the  experiments  of  succeeding  philosophers. 

As  the  constant  quantity  2  *^    is  the  same  as  that  al- 

ready found  for  capillary  tubes,  we  may  take  its  value,  namely, 
15,1311,  and  substitute  it  in  the  preceding  equation  ;  we  shall  then 
have 


and  since 


=  *'-T('-i). 


Capillary  Attraction.  347 

subtracting 

|(.-^)  =  0,1147, 

from  h  or  14,1544,  we  have 

h  =  14,0397, 

which   differs  0,4657   of  a  millimetre,  or  0,0183  of  an  inch,   from    457. 
13,574,  the  observed  altitude. 

460.  If  the   plates  are  inclined  to  each  other  at   a  small   angle, 

the  line  of  meeting  being  vertical,  the   water  will  rise  between  Fig.225. 
them   to   different   heights   according   to    the    general    law   above 
enunciated ;  that  is,  the  distances  at  L,   G,  being  LJV,   GI,  we 
shall  have 

GH  :  LM  :  :  LJV  :  GI  :  :  MO  :  HK. 

But,  by  similar  triangles, 

MO  :  HK  ::  FM  :  FH, 

whence 

GH  :  LM  ::  FM  :  FH, 

that  is,  the  heights  at  different  points  of  the  curve  ELGB  are 
inversely  as  the  distances  from  the  line  of  meeting  EF  of  the 
plates.  Therefore,  since  the  relation  of  the  lines  FM,  FH,  he, 
to  the  lines  LM,  GH,  he,  is  the  same  as  that  of  the  abscissas  to 
the  ordinates  in  the  common  hyperbola,  the  surface  of  the  fluid  be- 
tween the  plates  answers  to  this  curve. 

461.  If  the  relative  attraction  of  the  parts  of  the  fluid  for  itself, 

(in  the  case  of  tubes  for  example)  and  for  the  substance  of  the  Fig.  223 
tube,  be  such  that  the  surface  of  the  fluid  column  in  the  tube  be- 
comes convex,  instead  of  being  concave,  the  effect  is  precisely  the 
reverse  of  that  above  considered  ;  that  is,  when  an  equilibrium  oc- 
curs, the  filament  occupying  the  axis  of  the  tube  and  rising  without 
the  tube  to  the  free  surface,  will  have  its  extremity  F  depressed ; 
since,  instead  of  an  excess  of  upward  attraction  proceeding  from  a 
sustained  annulus,  situated  above  a  horizontal  plane  passing  through 
the  extremity  F,  there  will  be  a  deficiency  of  upward  attraction 
equal  to  the  effect  of  this  same  annulus.     Accordingly,  the  de- 


348  Hydrostatics. 

pressions  will,  like  the  elevations,  be  inversely  as  the  diameters  of 
the  tubes,  and  the  whole  ibeory  above  given,  with  this  single  modifi- 
cation, is  strictly  applicable.  Between  plates  also,  and  between 
concentric  tubes,  a  depression  will  take  place  corresponding  to  the 
elevation  in  the  case  where  the  upper  surface  is  concave. 

It  is  to  be  observed,  however,  that  the  deficiency  of  attrac- 
tion in  the  case  of  mercury  used  in  connection  with  glass,  taken 
in  its  ordinary  state,  is  to  be  ascribed  to  a  want  of  contact  be- 
tween the  fluid  and  the  substance  of  the  tube  or  plate,  arising 
from  a  film  of  moisture  which  ordinarily  attaches  itself  to  glass, 
and  which  being  completely  removed,  mercury  is  found  to  pre- 
sent a  concave  surface  like  water,  and  consequently  to  rise  in  a 
tube  and  between  plates  above  its  natural  level.*  Indeed  water 
may  be  made  to  exhibit  the  same  apparent  anomaly,  by  having  the 
surface  of  the  glass,  whether  tube  or  plate,  smeared  with  a  thin 
coat  of  tallow  or  wax. 

4G2.  The  peculiar  character  of  this  theory  consists  in  this, 
that  it  makes  every  thing  depend  upon  the  form  of  the  surface. 
The  nature  of  the  solid  body  and  that  of  the  fluid  determine  sim- 
ply the  direction  of  the  first  elements,  where  the  fluid  touches  the 
solid,  for  it  is  at  this  point  only  that  their  mutual  attraction  is 
sensibly  exerted.  These  directions  being  given,  they  become 
the  same  always  for  the  san;e  fluid  and  the  same  solid  substance, 
whatever  be  the  figure  of  the  body  itself  which  is  composed  of 
this  substance.  But  beyond  the  first  elements  and  beyond  the 
sphere  of  action  of  the  solid,  the  direction  of  the  elements  and 
the  form  of  the  surface  are  determined  simply  by  the  action  of  the 
fluid  upon  itself. 

We  have  seen  that  the  elevation  of  a  liquid  between  parallel 
plates,  is  half  of  that  which  takes  place  in  tubes  whose  diameter  is 

*  Barometer  tubes  properly  cleansed  and  freed  from  humidity, 
by  having  the  mercury  repeatedly  boiled  in  them,  will  exemplify 
the  truth  of  this  remark.  Moreover,  with  the  knowledge  of  this' 
fact,  we  can  readily  satisfy  ourselves  by  simple  inspection,  whether 
the  requisite  attention  was  paid  to  this  particular  in  the  construc- 
tion of  the  instrument. 


Apparent  Attraction  and  Repulsion  of  Floating  Bodies.     349 

equal  to  the  distance  of  the  plates.  The  cause  which  determines 
this  ratio  is  to  be  found  in  the  above  theory.  For,  in  the  case 
of  tubes,  the  action  of  the  concave  or  convex  surface  upon 
the  elevated  or  depressed  column  is  half  of  the  action  of  two 
spheres  which  have  for  radii  the  greatest  and  least  radii  of  the 
osculating  circles  to  the  surface  at  the  lowest  point.  The  tube 
being  flattened  in  any  direction,  the  radius  of  the  corresponding 
curvature  augments,  and  finally  becomes  infinite,  when  the  flat- 
tened sides  of  the  tube  become  parallel  plane  surfaces.  The 
first  part  of  the  attraction  of  the  surface  being  inversely  as  this 
radius,  will  become  zero,  and  there  will  remain  only  the  term 
depending  on  the  other  osculating  radius,  and  the  attractive  force 
is  accordingly  reduced  one  half.  Such  is  the  simple  and  rigor- 
ous result  furnished  by  the  theory  of  Laplace. 

463.  This  theory  serves  to  explain  also,  and  with  the  same 
simplicity,  all  other  capillary  phenomena.  Thus,  the  ascent  of 
water  between  concentric  tubes,  and  in  conical  tubes ;  the  curva- 
ture which  water  assumes  when  adhering  to  a  glass  plate ;  the 
spherical  form  observed  in  the  drops  of  liquids ;  the  motion  of  a 
drop  which  takes  place  between  plates  having  a  small  inclination 
to  each  other  and  to  the  horizon  ;  the  force  which  causes  drops 
floating  on  the  surface  of  a  liquid  to  unite  ;  the  adhesion  of  plates 
to  the  surface  of  a  liquid,  which  is  in  many  cases  so  great  as  to 
require  a  considerable  weight  to  separate  them  ;  —  these  effects,  so 
various,  are  all  deduced  from  the  same  formula,  not  in  a  vague 
and  conjectural  way,  but  with  numerical  exactness. 


On   the   apparent  Attraction  and   Repulsion   observed  in   Bodies 
jloating  near  each  other  on  the  Surface  of  Fluids. 

464.  (1.)  If  two  light  bodies,  capable  of  being  wetted,  be 
placed  at  the  distance  of  one  inch  from  each  other  on  the  surface 
of  a  basin  of  water,  they  will  float  at  rest,  and  without  approach- 
ing each  other.  But  if  they  be  placed  at  the  distance  of  only 
a  small  part  of  an  inch,  as  two  or  three  tenths,  they  will  rushFig.22& 
together  with  an  accelerated  motion. 


350  Hydrostatics. 

Fig.227.  (2.)  If  the  two  bodies  are  of  such  a  nature  as  not  to  suffer  the 
fluid  to  adhere  to  them,  as  is  the  case  with  balls  of  iron  used  in 
connection   with  mercury,  the  same  phenomena  will  be  observed. 

Fig.228.  (3.)  But  if  one  of  the  bodies  is  susceptible  of  an  adhesion  of 
the  fluid,  and  the  other  not,  as  two  balls  of  cork,  for  example, 
one  of  which  has  been  carbonized  by  the  flame  of  a  lamp ;  the 
effect  will  be  the  reverse  of  that  above  stated ;  that  is,  the  bodies 
will  seem  to  repel  each  other,  when  brought  very  near  together, 
and  with  forces  similar  to  those  whh  which  in  the  former  ease 
they  tended  to  unite. 

Moreover,  a  single  ball  will  approach  to,  or  recede  from,  the 
side  of  the  vessel,  as  it  would  approach  to  or  recede  from  another 
ball,  according  as  the  substance  of  the  vessel  and  that  of  the 
ball  are  similar  or  dissimilar  as  to  their  disposition  to  cause  an 
adhesion  of  the  fluid. 

465.  In  these  experiments  the  approach  and  recession  of 
the  floating  bodies  are  not  the  effect  of  a  real  attraction  or  repul- 
sion between  the  bodies  ;  for,  if  the  bodies,  instead  of  being  plac- 
ed upon  the  surface  of  a  liquid,  be  suspended  by  long,  slender 
threads,  nothing  of  the  kind  is  to  be  perceived.  We  must  there- 
fore look  for  some  other  cause  to  which  to  refer  these  appearan- 
ces. 

Fig.229.  If  two  plates  of  glass  AB,  CD,  be  suspended  in  water  paral- 
lel to  each  other,  and  at  such  a  distance,  that  the  point  H,  where 
the  two  curves  of  elevated  fluid  meet,  shall  be  on  a  level  with 
the  common  surface,  the  plates  will  remain  in  equilibrium.  But 
on  being  brought  so  near  to  each  other,  that  the  point  H  shall 
be  above  the  common  level  of  the  surface,  the  mass  of  fluid  thus 
raised  will  have  the  effect  of  a  chain  attached  at  its  extremities 
to  the  plates,  in  drawing  the  plates  toward  each  other.  The 
approach  of  the  balls  to  each  other  under  similiar  circumstances, 
is  to  be  referred  to  the  same  cause. 

When  the  point  H  is  below  the  general  level,  on  account  of 
a  want  of  adhesion  in  the  parts  of  the  fluid  to  the  plates,  the 
pressure  of  the  plates  inward  toward  each  other  not  being  coun- 
terbalanced by  the  pressure  in  the  opposite  direction,  they  must 
approach  each  other,    and  with  a  greater  or  less  force,  according 


Barometer.  351 

to  the  depression  of  the  point  iJ,  or  the  nearness  of  the  plates  to 
each  other.  This  affords  an  explanation  of  the  second  case  above 
stated. 

If  one  of  the  floating  bodies,  as  A,  for  example,  is  susceptible  Fig. 
of  being  wetted,  while  the  other  B  is  not,  the  fluid  will  rise  around  A 
and  be  depressed  around  B.  Accordingly,  when  the  balls  are  near 
to  each  other,  the  depression  around  B  will  not  be  symmetrical,  and 
the  body  being  thus  placed  as  it  were  upon  an  inclined  plane,  its 
equilibrium  will  be  destroyed,  and  it  will  move  off  from  the  other 
body  in  the  direction  of  the  least  pressure. 

These  phenomena,  of  which  we  have  given  only  a  familiar 
explanation,  are  all  comprehended  in  Laplace's  theory  of  ca- 
pillary attraction  ;  and  the  attractive  and  repulsive  forces  are  capa- 
ble, on  that  theory,  of  being  subjected  to  a  rigorous  calculation. 

Of  the  Barometer. 

466.  If  we  take  a  glass  tube  thirty-three  or  thirty-four  inches 
in  length,  closed  at  one  extremity  and  open  at  the  other,  and 
having  filled  it  with  mercury,  place  the  finger  over  the  open  ex- 
tremity and  thus  immerse  it  in  a  basin  of  the  same  liquid  with- 
out suffering  the  air  to  enter  the  tube,  the  mercury  will  settle 
down  in  the  tube,  leaving  a  vacuum  above  it,  till  its  weight  is 
exactly  counterbalanced  by  the  pressure  of  the  atmosphere,  ex- 
erted '  ^^on  the  surface  of  the  mercury  in  the  basin.  This  instru- 
ment .  r  called  a  barometer.*  The  perpendicular  height  at  which 
the  mercury  is  ordinarily  maintained  at  the  level  of  the  sea,  is  very 
nearly  thirty  inches. 

From  what  has  been  said  of  the  manner  in  which  the  pres- 
sure of  fluids  is  propagated,  it  will  be  perceived  that  it  is  imma- 
terial what  be  the  extent  of  surface  in  the  basin,  or  whether  the 
atmospheric  pressure  be  applied  at  the  top,  or,  by  means  of  a 
flexible  bag  containing  the  liquid,  at  the  bottom  and  sides.  If 
instead  of  entering  a  basin  the  tube  turn  up  at  'the  bottom,  as  in 
figure  230,  so  as  to  admit  the  air  at  C,  the  perpendicular  elevation 

*  See  note  on  the  construction  and  history  of  the  barometer. 


362  Hydrostatics. 

above  a  horizontal  line  coinciding  with  the  surface  at  E  or  jF, 
will  be  the  measure  of  the  atmospheric  pressure.  This  eleva- 
tion, moreover,  is  independent  of  the  form  of  the  tube  and  the 
particular  quantity  of  mercury  contained  in  it.  On  the  suppo- 
Fig.  232.  sition,  however,  that  the  base  of  the  tube  is  an  inch  square,  the 
pressure  is  equal  to  that  of  a  parallelopiped  of  mercury  30  inches 
in  length,  or,  which  amounts  to  the  same  thing,  to  the  weight  of 
410.  30  cubic  inches  of  mercury.  Now  30  cubic  inches  of  water  is 
equal  to  30  X  252,525  grains,  or  15,78  troy  ounces.  Whence 
30  cubic  inches  of  mercury  is  equal  to  15,78  X  13,57  or  214,12 
troy  ounces;  that  is,  to  234,7  ounces  avoirdupois,  or  to  14,7'^'. 
We  infer,  therefore,  that  the  pressure  of  the  atmosphere  amounts 
to  nearly  15'^-  upon  every  square  inch  of  surface,  or  to  about 
one  ton  upon  every  square  foot.  A  common  sized  man  exposes 
a  surface  of  10  or  11  feet,  and  is  consequently  subjected  to  a  pres- 
sure of  as  many  tons'  weight.  The  entire  surface  of  the  earth 
being  estimated  at  5575680000000000  feet,  this  number  will  ex- 
press the  weight  nearly  of  the  whole  atmosphere  in  tons,  a  certain 
deduction  being  made  for  the  space  occupied  by  mountains  and  ele- 
vated regions.  This  pressure  being  exerted  upon  the  surface  of 
the  ocean,  fishes  are  exposed  to  it  in  addition  to  the  weight  of  their 
natural  element.     But  the  proportion 

1    :  13,57  :  :  30  :  407,1, 

gives  407,1  inches,  or  34  feet  nearly,  for  the  length  of  a  column 
of  water  equivalent  to  that  of  30  inches  of  mercury  or  the  pres- 
sure of  the  atmosphere.*  Accordingly  for  every  34  feet  depth 
a  pressure  is  exerted  of  a  ton  upon  every  foot  of  surface,  over 
and  above  that  arising  from  the  atmosphere.  Now  fishes  are 
sometimes  caught  at  the  depth  of  2600  or  2700  feet,  where  the 
pressure  of  the  water  amounts  to  nearly  80  atmospheres  or  80 
tons  upon  a  square  foot ;  yet  these  fishes  are  not  injured  by  such 
an  immense  weight,  or  sensibly  impeded  in  their  motions.  The 
reason  is,  that  they  are  filled  with  fluids,  which  from  their  im- 
penetrability oppose  a  sufficient  resistance  to  this  pressure,  and 
thus  preserve  the  most  delicate  membranes  from  being  ruptured. 

*  The  mean  pressure  of  the  atmosphere  is  more  accurately 
estimated  at  29,922  English  inches. 


Barometer.  353 

With  regard  to  the  facility  and  rapidity  of  their  motions,  as  the 
incumbent  weight  acts  equally  in  all  directions,  it  neutralizes 
itself,  by  aiding  just  as  much  as  it  obstructs  their  efforts  to  move 
and  turn  themselves.  The  case  is  precisely  similar  with  respect 
to  land  animals.  The  vessels  of  the  animal,  together  with  the 
bones,  are  filled  with  air  or  some  other  fluid  capable  of  support- 
ing any  weight,  and  whose  elasticity  being  equal  to  that  from 
without,  proves  an  exact  counterbalance  to  it. 

467.  In  the  barometer  there  is  an  equilibrium  between  the 
pressure  of  the  mercury  and  that  of  the  atmosphere.  Now  we 
have  seen  that  when  two  fluids  thus  counteibalance  each  other, 
the  altitudes  must  be  inversely  as  the  specific  gravities.  Accord-  411. 
ingly  as  the  specific  gravity  of  mercury  is  to  tliat  of  air  at  the 
surface  of  the  earth  as  13,57  to  0,00122,  we  shall  have 

0,00122  :  13,57  :  :  30  :  "^^^^-^  —  333688. 

We  infer,  therefore,  that  the  height  of  the  atmosphere,  on  the 
supposition  of  a  uniform  density  throughout,  is  333688  inches, 
or  a  litde  more  than  5  miles.  But  the  air  being  eminently  elas- 
tic, the  lower  strata  are  compressed  by  the  incumbent  weight  of 
those  above,  so  that  the  density  becomes  less  and  less  continu- 
ally as  we  ascend.  Let  the  weight  of  the  column  of  mercury 
which  measures  the  pressure  of  the  atmosphere,  exerted  upon  a 
unit  of  surface,  be  denoted  by  o-  a  A,  g  being  the  force  of  gravity, 
A  the  density  of  the  mercury,  and  h  the  perpendicular  height  of 
the  column  above  the  level  of  the  surface  in  the  basin,  and  let 
the  weight  of  the  atmosphere  upon  the  same  surface  be  denoted 
by  w,  we  shall  have 

g  A  h  =:  tv. 

As  we  ascend  into  the  atmosphere,  die  weight  w  and  the  height 
h  diminish  continually,  and  these  diminutions  depend  upon  the 
elevation  attained,  and  the  law  according  to  which  the  densities 
of  the  atmospheric  strata  decrease.  If  this  law  were  known,  it 
might  be  made  use  of  for  the  purpose  of  determining  the  difTer- 
ence  in  the  altitudes  of  two  points  above  a  common  level,  as  the 
sea,  or  any  assumed  level.  But  in  order  to  discover  this  law,  it 
Mech.  45 


354  Hydrostatics. 

is  necessary  to  recur  to  certain  experiments  relating  to  the  den- 
sity of  the  air  under  different  pressures  and  at  different  tempe- 
ratures. 

Fig.  L30.  468.  Take  a  recurved  glass  tube  JIBC^  open  at  the  extremity 
A  and  closed  at  the  other  extremity  C  ;  pour  into  it  a  quantity 
of  mercury  just  sufficient  to  fill  the  bended  part  up  to  the  hori- 
zontal line  DE,  so  that  the  air  confined  in  the  shorter  branch 
CE  may  be  ncitiier  more  nor  less  pressed  than  that  contained 
in  the  longer  branch  AD^  which  communicates  with  the  atmo- 
sphere. The  mercury  being  at  the  same  height,  therefore,  in 
each  branch,  and  the  communication  with  the  external  air  being 
cut  off,  if  we  introduce,  by  means  of  a  fine  tunnel,  more  mercury, 
we  shall  observe  this  liquid  to  stand  higher  in  the  branch  BA 
than  in  the  other,  whereby  the  air  in  EC  will  be  condensed, 
the  compressing  force  being  equal  to  the  difference  of  the  two 
columns.  If  tlie  space  EC,  supposed,  for  example,  to  be  4  inch- 
es, were  reduced  one  half  or  to  F,  by  the  pressure  of  a  column 
of  mercury  extending  to  H,  drawing  the  horizontal  line  FG,  we 
should  find  the  difference  GiJof  thetwo  columns  exactly  equal 
to  the  height  of  the  barometer  at  the  time  of  the  observation  ;  so 
that  the  air  contained  in  the  space  CF  would  be  pressed  by  the 
weight  of  the  atmosphere  incumbent  upon  H  and  by  the  weight 
of  another  atmosphere  represented  by  the  column  GH.  A  double 
pressure,  therefore,  reduces  the  bulk  one  half.  If  we  continue 
to  add  to  the  weight  by  pouring  in  more  mercury  till  the 
confined  air  is  condensed  to  F'  or  to  one  third  of  the  original 
space,  we  shall  find  the  additional  quanUty  necessary  to  this 
effect  the  same  as  before,  that  is,  the  column  GH'  will  be 
equivalent  to  two  atmospheres.  Thus  a  triple  pressure  reduces 
the  bulk  of  the  confined  air  to  one  third  the  space.  We  might 
continue  to  increase  the  weight,  and  we  should  in  every  instance 
obtain  results  agreeable  to  the  same  jreneral  law. 

So,  on  the  other  hand,  by  diminishing  the  natural  pressure 
exerted  upon  any  portion  of  air,  we  shall  still  find  the  bulk  in- 
versely proportional  to  the  pressure.  Let  the  tube  ABC  be 
supposed  to  have  a  bore  not  exceeding  one  tenth  of  an  inch. 
A  drop  of  mercury  being  introduced  at  the  bend  A,  if  the  whole 
apparatus  be  placed  under  the  receiver  of  an  air-pump,  and  the 


Barometer.  355 

air  be  exhausted  from  the  longer  branch,  till  the  pressure  is  re- 
duced successively  one  half,  two  thirds,  &c.,  the  portion  of  air 
confined  by  the  drop  of  mercury  will  expand,  driving  the  drop 
before  it,  and  will  occupy  successively,  double,  triple,  &c.,  of  its 
original  bulk.  We  infer,  therefore,  universally,  that  the  space 
occupied  by  any  given  portion  of  air  is  reciprocally  proportional  to 
the  pressure. 

In  order  that  this  law  may  hold  true,  however,  in  the  strictest 
sense,  it  is  to  be  remarked  that  the  air  must  be  perfectly  dry ; 
for  the  small  quantity  of  aqueous  vapor,  which  is  ordinarily  found 
mixed  with  the  atmosphere,  is  not  condensed  by  pressure  accord- 
ing to  the  same  law,  as  will  be  shown  hereafter. 

The  instrument  represented  by  figure  230,  is  called  a  manome- 
ter. It  is  used  for  the  purpose  of  measuring  the  elastic  force  of  other 
gases  besides  the  atmosphere  ;  and  they  are  all  found  to  be  con- 
densed and  expanded  according  to  the  above  law.  This  impor- 
tant property  was  discovered  by  JMariotte,  and  is  frequently  referred 
to  under  the  name  of  the  law  of  Mariotte. 

4G9.  Recurring  to  the  first  experiment  above  described,  the 
pressure  exerted  upon  the  portion  £C  of  confined  air,  when  the 
recurved  part  of  the  tube  is  just  filled  with  mercury,  is  that  of 
the  atmosphere,  or  ^  A  h.  But  this  pressure  is  resisted  and 
counterbalanced  by  the  elasticity  of  the  confined  air,  which  by 
supposition  is  of  the  same  density  with  that  immediately  sur- 
rounding the  apparatus.  We  may  take  g  ^  h,  therefore,  as  the 
measure  of  the  elastic  force  of  the  air  in  question.  This  force 
remains  the  same  so  long  as  the  air  continues  of  the  same  density 
and  the  same  temperature.  If  a  manometer  be  removed  from 
one  place  to  another,  care  being  taken  not  to  change  the  state  of 
the  confined  air,  the  product  g  A  h  which  represents  the  elastic 
force  does  not  undergo  any  change.  But  if  the  gravity  g  varies 
as  we  remove  from  one  place  to  another,  the  height  h  of  the 
mercury  will  also  vary  in  the  inverse  proportion  to  g,  the  density 
A  of  the  fluid  being  supposed  to  remain  the  same.  It  will  hence 
be  perceived  that  the  variations  in  the  heights  of  the  mercury  in 
the  manometer  are  capable  of  rendering  sensible  the  variations 
of  gravity,  and  may  even  be  employed  in  determining  the  augmen- 
tations or  diminutions  of  this  force  arising  from  changes  of  distance 
with  respect  to  the  centre  of  the  earth. 


356  Hydrostatics. 

470.  Let  us  now  suppose  that,  the  weight  of  the  atmosphere 
remaining  the  same,  the  temperature  of  the  confined  air  is  rais- 
ed ;  as  this  air  expands,  its  bulk  will  be  increased,  and  its  density 
diminished.  Now  we  know  by  the  careful  experiments  of  M.  Gay 
Lussac  and  others,  (1.)  That  all  the  gases  dilate  uniformly, 
at  least  from  32°  to  212°,  or  from  the  freezing  to  the  boiling  point 
of  water.  (2.)  That  the  dilatation  arising  from  the  same  increase 
of  heat  is  precisely  the  same  for  all  the  gases,  vapors,  and 
mixtures  of  gases  and  vapors.  (3.)  That  the  bulk  of  confined 
gas,  at  the  temperature  of  32°,  being  considered  as  unity,  this 
common  dilatation  is  0,375,  (or  a  little  more  than  one  third,) 
for  180°,  the  difference  between  the  boiling  and  freezing  points 
of  water ;  which  gives  °j\Y  =  t lo  OJ"  0.00208  for  the  augmen- 
tation of  bulk  answering  to  1°  of  Fahrenheit.  Accordingly,  we 
shall  have  for  the  bulk  or  space  occupied  by  the  portion  of  air 
in  question  1  -\-  0,00208  n  at  the  temperature  denoted  by  n,  the 
number  of  degrees  above  or  below  32,  the  latter  being  considered 
as  negative.  This  bulk  or  volume  may  be  reduced  to  its  original 
limits,  by  bringing  the  temperature  back  to  32°,  or  by  increasing 
or  diminishing  the  weight  which  compresses  it,  without  alter- 
ing the  temperature.  It  would  only  be  necessary,  in  this  latter 
case,  to  add  to  or  take  from  the  weight  w  a  portion  equal  to 
t(;(0,00208)  n,  that  is,  to  substitute  for  2v  the  weight 

u;  (1  +  0,00208  n), 

which  is  the  measure  of  the  elastic  force  of  the  confined  air  re- 
duced to  its  original  density.  Hence,  the  bulk  and  density  re- 
maining the  same,  the  elastic  force  varies  with  the  temperature,  and 
in  the  same  ratio. 

If  the  elastic  force  is  proportional  to  the  density  when  the 
temperature  is  ihe  same,  antl  varies  with  the  temperature  when 
the  density  is  the  same,  it  will  be  easy  to  deduce  tiie  value  of 
this  force  in  terms  of  the  two  elements,  on  the  supposition  that 
they  both  vary  together.  Thus,  putting  A  for  the  density  of  the 
air  in  question,  and  n  for  the  number  of  degrees  which  marks  the 
temperature,  and  p  for  its  elastic  force  or  pressure  exerted  upon 
the  unit  of  surface,  a  being  the  ratio  of  the  elastic  force  to  the 
density  at  the  temperature  of  32°,  we  shall  have 


Barometer  applied  to  the  Measurement  of  Heights.        357 

p  =  «  A  (1  +  0,00208  »).  (i.) 

The  coefficient  a  is  constant  for  the  same  elastic  fluid,  but  is  dif- 
ferent in  different  fluids,  and  requires  to  be  determined  in  each 
particular  case. 

471.  In  applying  the  results  above  stated  to  the  mass  of  air 
which  composes  the  atmosphere,  we  take  into  consideration  only 
a  single  vertical  column  of  air,  supposed  lo  rest  upon  the  surface 
of  the  earth  and  to  extend  indefinitely  upward.  We  may  con- 
ceive of  the  surrounding  mass  or  atmosphere  as  congealed  or  ren- 
dered solid.  If  it  were  previously  in  a  state  of  equilibrium,  this 
state  will  not  be  disturbed  by  such  a  supposition ;  so  that  the 
column  in  question  will  still  be  in  equilibrium  as  before.  Now 
the  force  which  acts  upon  the  particles  of  air  is  gravity,  which 
may,  without  sensible  error,  be  regarded  as  exerting  itself  in 
the  direction  of  the  aerial  column  throughout  its  whole  extent, 
or  at  least  as  far  as  it  is  necessary  to  take  any  account  of  it.  Ac- 
cordingly, it  is  necessary,  in  order  to  an  equilibrium,  that  the  den- 
sity, the  pressure,  and  the  temperature  should  be  considered  as 
uniform  throughout  a  horizontal  stratum  of  infinitely  small  thick- 
ness. The  column  being  composed  of  an  infinite  number  of 
these  strata  or  lamina,  let  h  be  the  height  or  distance  from  the 
surface  of  the  earth  of  one  of  these  strata,  A  the  density  of  this 
stratum,  t  its  temperature,  g'  its  gravity,  p  its  elastic  force,  a  its 
base,  and  d  h  its  thickness.  We  shall  have  a  p  for  the  pressure 
exerted  upon  the  inferior  base,  and  a  (p  —  d  p)  for  the  pressure 
upon  the  superior  base  ;  the  difference  —  a  d  p  must  be  equal  to 
the  weight  a  a  g'  d  h  of  this  stratum.  Hence,  by  suppressing 
the  common  factor  a,  we  have  the  equation 

—  d  p  z=z  A  g'  d  h, 

or,  substituting  for  a  its  value     ,.    , r   deduced   from   equation 

^  a  {]  -{■  r  7i)  ^ 

(i.),  the  fraction  0,00208  being  for  the  sake  of  brevity  represented 
by  e. 


d  p  =      ,.    .  g'  d  h. 


Whence 


358  Hydrostatics. 

dp  —  g'  d  h 

p  a  (1  +  e  n)' 

Nothing  can  be  inferred  from  this  equation  until  the  value  of  n 
is  given  in  terms  of  h.  Now  we  know  that  the  temperature  de- 
creases as  we  ascend  from  the  surface  of  the  earth,  but  the  law 
of  this  decrease  has  not  been  determined  in  a  manner  altogether 
satisfactory.  Fortunately,  this  law  has  little  influence  upon  our 
results  in  the  calculation  of  heights  by  the  barometer,  on  ac- 
count of  the  smallness  of  the  coefficient  e ;  and  we  may,  in  ques- 
tions of  this  kind,  consider  the  temperature  as  constant,  provided 
we  take  for  n,  in  each  particular  case,  the  mean  of  the  tempera- 
tures observed  at  the  two  extreme  points  of  the  height  h  to  be 
determined.  Moreover,  r  being  the  radius  of  the  earth,  and  g 
the  gravity  at  the  surface,  we  have,  at  the   distance  v.  -\-  h  from 

the  centre, 

g  K^ 

6     —  (r  -f  A)2' 

since  this  force  varies  in  the  inverse  ratio  of  the  square  of  the 
distance.  The  preceding  equation  becomes,  by  this  substitu- 
tion, 

dp  —  g  K^  d  h 

~Y  ~  a  (1  -{-  e  n)  (r  -1-  A)2* 

Whence,  by  integrating  on  the  supposition  that  n  is  constant,  we 
have 

Cal.  112.  w  being  equal  to  0,434295,  log.  denotes  the  common  logarithm  of 
p.  To  determine  the  constant  C,  let  w  be  the  value  of  p  answer- 
ing to  ^  =  0  ;  and  we  shall  have 


,  m  g  R  ,     r^ 

log.  vj  =  —pi-l \  +  ^^ 

^  a  (1  4-  e  n)    ' 


(1+en) 

Consequently,  by   subtracting   the   preceding   equation   from  this, 

we  obtain 

,cy  m  g  R              h  ,     . 

^S'  p  ~  a(l-f  en)  •  5r+^*  ^"•'' 

This   equation,    taken   in  connection  with   equation  (r.),   gives  the 

values  of  p  and  A  in  terms  of  h.     Thus  we  have  equations  con- 


Barometer  applied  to  the  Measurement  of  Heights.        359 

taining  the  laws  of  the  density  and  elastic  force  of  the  air  which 

belong  to  a  state  of  equilibrium  in  the  atmosphere. 

472.  To  make  use  of  equation  (ii.)  for  the  purpose  of  measur- 
ing heights  by  means  of  the  barometer,  let  us  suppose  the  barome- 
tric altitude  at  the  surface  of  the  earth  and  at  the  height  h  to  be 
known  by  actual  observation,  and  let  them  be  denoted  respectively 
by  IV,  w',  the  corresponding  temperatures  of  the  mercurial  columns 
being  represented  by  t,  t'.  The  expansion  of  mercury  being  g^V^ 
or  0,0001025,  that  is,  0,0001  nearly,  for  each  degree  of  Fahrenheit's 
scale,  if  d  be  the  density  corresponding  to  the  temperature  t  of  the 
mercury  at  the  first  station, 

D  (I  4-  0,0001)  (t  —  t') 

will  be  the  density  which  answers  to  the  temperature  of  the  mer- 
cury at  the  second  station.     Accordingly  we  have 

a  =  D  g  w,     and     p  =  n  g'  w'  (1  -f-  0,0001)  (t  —  t'). 

The  correction  for  the  upper  barometric  column  on  account  of 
difference  of  temperature  being  made  agreeably  to  this  formula,  we 
may  consider  w'  as  representing  the  length  of  this  column  thus 
corrected.  Whence,  dividing  the  first  of  the  above  equations  by 
the  second,  we  have 

„^^^.     (M^»   (,„.) 

J)  g'  iv'  to'  R-^  ^         ' 

substituting  for  g'  its  value  7^-^7—7-^2  >  ^"^  consequently, 

log.  J  =  log.  ^  +  2  log.  (1  +  ^*),        (ni.) 


smce 


(r  +  702 


=C4^>=0+D^- 


Let  T,  t',  be  the  temperatures  respectively  of  the  air  at  the 
surface  of  the  earth  and  at  the  height  h ;  t,  t',  will  generally 
differ  from  t,  t',  the  temperatures  of  the  mercury  in  the  barome- 
ter, since  the  latter  is  not  ordinarily  allowed  sufficient  time  to 
acquire    the    temperature   of    the   surrounding  air.     t,  t',  are    to 


360  Hydrostatics. 

be  taken  by  means  of  a  thermometer  suspended  in  the  air, 
while  T,  t',  arc  supposed  to  be  indicated  by  a  thermometer  at- 
tached to  the  barometer.   We  take  ?t  =       ^ 32.      IMoreover, 

the  coefficient  g-i^  or  0,0020S,  rcjjresenting  the  elastic  force,  re- 
quires to  be  increased  somewhat  for  the  purpose  of  taking  ac- 
count, as  far  as  can  be  done,  of  the  quantity  of  water  in  a  state 
of  va[)or  which  is  at  all  times  mixed  with  the  air  in  a  greater 
or  less  quantity.  Indeed,  under  the  ordinary  pressure  of  the  at- 
mosphere, the  density  of  aqueous  vapor  is  to  that  of  air,  as  10 
to  14  ;  consequently,  the  atmosphere  is  so  much  the  lighter  accord- 
ing as  it  is  composed  in  a  greater  degree  of  this  vapor.  Now  it 
contains  so  much  the  more  vapor  according  as  its  temperature  is 
more  raised,  whereby,  when  the  air  is  dilated  by  heat,  its  weight 
must  be  diminished  in  a  higher  ratio  than  that  of  its  augmentation 
of  bulk.  We  increase  the  coefficient  0,00208  dierefore  to  0,00223 
or  ^iy,  which  has  been  found  by  actual  trial  to  give  the  most  cor- 
rect resuhs.     We  have,  accordingly, 

e,  n  =.0,00223  Q-^^  —  32<^\ 

We  now  substitute  in  equation  (ii.)  for  e  n  the  above  value, 
and  for  log.  -  the  value  found  in  equation  (ni.),  and  we  shall  ob- 
tain 

log.  -,  +  2  log.  (1  H-  'A  = '^^^, ^  X  -47- 

^  "  -  ^     ^  k;       ^  ^j  ^  0..00223(:L+Z:_32o))        ^  +  ^' 


Whe 


nee 


^  =  log.^,  +  2^ 


a  (1  +  0,00223  (^±^  -  32°))  (r  +  h) 
5(1  +  -) ^^-^= — 

=  ;^  (l  +  0,00223  (I±l  _  32-))  bS.  ;^  +  2  log.  (I  +  ^)  (1  + *)  (.V.) 
The  best   means  of  determining  the  coefficient  - —  of   this 


Barometer  applied  to  the  Measurement  of  Heights.       361 

formula,  is  to  make  use  of  a  height  (or  rather  a  number  of 
heights),  well  known  by  actual  measurement,  or  by  trigonomet- 
rical operations.  We  then  substitute  for  A  this  known  value, 
and  for  iv,  w',  t,  t',  the  lengths  of  the  barometrical  columns,  and 
the  temperature  of  the  air  at  the  two  stations  respectively,  and 
for  R  the  mean  radius  of  the  earth,  namely  34S12S0  fathoms.     We 

shall  thus   have    an    equation  from  which  the  value  of is  read- 

'■  m  g 

ily  deduced  once  for  all.  Taking  the  mean  result  of  a  great 
number  of  observations  conducted  with  the  greatest  care,  by   M. 

Ramond,   we  find  —  ,  for  the  latitude  45°,  *   equal   to    1S33G  t 
m  g  '         n 

metres,  or  10026  English  fathoms.  This  is  on  the  supposition 
of  a  temperature  of  32°,  and  agreeably  to  what  has  been  said,  it 
may  be  increased  or  diminished  by  adding  or  subtracting  y^^ 
or  a  0,00223  part  for  each  degree  above  or  below  32°.  We 
can  therefore  reduce  it  to  10000,  instead  of  10026,  by  supposing 
the  temperature  somewhat  lower.  Thus,  since  26  is  0,0020  of 
10000 

0,00223  :  0,0026  :  :  1°  :   1°,16. 

If,  therefore,  we  subtract  1°,16  from  32°,  we  shall  have  30°, 84 
or  31°  nearly,  for  the  temperature  at  which  the  constant  coeffi- 
cient is  10000  fathoms. 

473.  Since  this  coefficient  contains  g,   it   must    vary  with   g, 
that  is,  with  the  latitude.     Now,  according  to  the   law  of  the  va- 


*  This  coefficient  was  actually  determined  for  the  latitude  of 
about  43°.  But  the  correction  for  small  distances  in  latitude  is  so 
inconsiderable,  that  it  may  be  regarded  as  nothing.  Moreover, 
the  coefficient,  if  corrected  at  all,  would  require  to  be  diminished, 
and  it  is  thought  on  the  whole  less  liable  to  error  by  excess  than  by 
deficiency. 

t  The  coefficient  deduced  theoretically  from  the  relative  densi- 
ties of  mercury  and  air,  as  determined  by  Biot  and   Arago,   allow- 
ance   being  made  for  humidity,    is   18334,1    metres,   differing  less 
than  2  metres,  that  is,  less  than  2  X  39,371  inches,  from  the  above. 
Mech.  46 


362  Hydrostatics. 

riation  of  gravity  in  different  latitudes,  if  g  represent  the  value 
of  this  force  in  the  latitude  of  45°,  and  g'  that  of  any  other  lati- 
tude L,  we  shall  have 

g^  —g  (I  —  0,002837  cos  2  L).  * 

At  45*^,  therefore,  where  cos  2  L  =:  0,  g'  =  g,  or  the  correction, 
is  0  ;  and  for  higher  latitudes  the  correction  is  — ,  or  subtractive, 
and   for  lower  latitudes  it   is  +  or  additive.     Whence,  generally, 

—  =  lOOOO^''"'-  (1  +  0,002837  cos.  2  L.) 
m  g  \      I      J  J 

By  means  of  this  value  of ,    substituted    in    equation    (iv.) 

*  The  value  of  g'  in  different  latitudes  depends  upon  the  par- 
ticular figure  of  the  terrestrial  spheroid,  the  determination  of  which 
belongs  to  astronomy.  We  will  merely  observe  in  this  place,  that  a 
comparison  of  articles  346,  347,  conducts  us  directly  to  the  equation 

£  —  ± 
g  «' 

a,  a',  being  the  lengths  of  the  pendulum  corresponding  to  the  parts 
of  the  earth  in  which  the  intensities  of  gravity  are  g,  g' ,  respective- 
ly. Now  it  is  found  that  the  general  expression  for  the  length  of 
the  seconds  pendulum,  the  day  being  divided  into  100000  seconds,  is 

metre.  metre. 

a  =  0,739502  +  0,004208  (sin  L)^. 
Hence,  since 

(sin  i)2  =  J  (1  —  cos  2  L),  and  (sin  45°)2  =  |, 
Trig.  20.  we  shall  have 
'^"^-  ^^-  ±  _  g__  0,739502  +  0,002104 

a!    ~  ^~  0,739502  +  0,004208  (sin  Lf ' 


0,002837  cos  2  i  ' 


therefore, 

g  =g  {y—  0,002837  cos  2  L). 

In  the  original  memoir  of  M.  Ramond,  the  coefficient  stood 
0,002845,  and  it  was  thus  copied  by  Laplace  and  others.  It  was 
afterward  corrected  by  M.  Oltmanus,  and  the  error  acknowledged 
by  the  author  in  a  separate  edition  of  the  memoir. 


Barometer  applied  to  the  Measurement  of  Heights.       363 

we  shall  be  able  to  determine  the  height  h  in  any  part  of  the 
earth,  when  iv,  w',  t,  r',  corresponding  to  the  extreme  points  of  h, 
are  known  ;  or  we  may  retain  the  coefficient  10000  unaltered, 
and  apply  the  correction  to  the  result,  according  to  the  following 
table ; 


Latitude. 

Correction. 

0° 

+  3T2"  ^^  ^^  approximate  height 

5° 

.      +t1b 

lOo 

•           +7^ 

15° 

•      +T^ 

20° 

"T     46^10 

25° 

•      +   T4-g- 

30° 

.            +T^ 

35° 

•      +    ToTo 

40° 

+    2  oV^ 

45° 

.     +    0 

50° 

2'0  3  0 

55° 

1 

•                   1  030 

60° 

7UT 

65° 

1 

5  48 

70° 

1 

460 

75° 

•        40  7 

80° 

1 

¥7T 

85° 

1 
3TT 

90° 

3T^ 

474.  As  the  fraction  -   is   always  very   small,  we    shall   have 

very  nearly  the  value  of  h,  independently  of  the  term  containing 
this  fraction  j  by  substituting  the  approximate   value  thus  obtained 

for  A,  in  the  fraction  -,  we  shall  have  very  nearly  the  correction 

due  to  the  variation  of  gravity  at  different  elevations  in  the  same 
latitude  ;  and  by  substituting  the  value  of  h  thus  corrected  in  the 

fraction  -  we  can  approximate  the  true  height  still  more  nearly. 

But  this  second  substitution  is  altogether  superfluous  in  the  cases 
which  ordinarily  occur.      Indeed,  except  where  h  is  very  great, 

we  may  neglect  -  entirely,  and   the  general  formula  then  becomes 

R 


364  Hydrostatics. 

''  =  '-^  ('  + "'"'"''  (^)  -  '^0 '°°-  ^'- 

It  will  be  perceived  that may   require    some    modification^ 

in  order  that  the  formula  in  this  state  should  adapt  itself  to  ob- 
served heights  or  known  values  of  h ;  and  indeed  the  observa- 
tions of  ]M.  Ramond   give,   for  the  value  of  the  coefficient  to  be 

employed   in   this   formula,  =   1S393  metres  or  10031   fath- 

I     ■'  m  g 

oms,  exceeding  the  former  by  5  fathoms.  Accordingly  the  de- 
pression of  the  temperature  below  32^,  required  in  order  to 
change  this  to  the  more  convenient  form  of  10000,  will  be  found 
to  be  1°,45  ;  retaining  the  coefficient  10000,  therefore,  we  have 
only  to  suppose  the  temperature  32°  —  1°,45  or  30°, 55.  As 
this  differs  less  than  half  a  degree  from  31°,  and  as  we  can  sel- 
dom be  certain  oi  the  temperature  of  the  air  to  a  greater  degree 
of  accuracy,  we  may  still  use  the  same  formula  without  any  other 

change  than  the  omission  of  the  term  depending  on  -  .     We  have 

hence  a  very  simple,  convenient,  and,  for  common  cases,  sufficiently 
exact  formula,  namely, 

h  =  10000  (1  +  0,00223  ('^t^)  —  31°)  (log.  t«  —  log.  w'). 

This  being  adapted  to  the  latitude  of  45°,  when  the  barome- 
trical observations  relate  to  a  place  on  a  parallel  considerably 
distant  either  north  or  south,  it  will  be  seen  directly  by  the  fore- 
going table  when  it  is  necessary  to  apply  a  correction  for  differ- 
ence of  latitude,  and  what  tliis  correction  is.  It  will  be  recol- 
lected that  the  lengths  of  the  barometric  columns  w,  w',  which 
represent  the  weights  of  the  atmosphere  respectively  at  the  two ' 
stations,  are  supposed  to  be  reduced  to  the  same  temperature. 
The  upper  column  w'  is  usually  the  coldest,  and  consequently 
too  short.  Now,  according  to  the  rate  of  expansion  or  contrac- 
tion already  mentioned,  as  1  inch  is  shortened  0,0001  for  each 
degree,  a  column  of  25  inches  will  be  shortened  0,0025  of  an 
inch,  for  each  degree  of  depression,  and  consequently  0,01  for 
evory  —  4°  ;  and  each  portion  of  2,5  inclies  will  be  shortened 
oue  tenih  part  of  this,  or  0,001   for  the  same  amount  of  depres- 


Barometer  applied  to  the  Measurement  of  Heights.         365 

sion  or  —  4°.  In  common  chamber  barometers,  the  lengths 
of  the  columns  are  read  off  to  a  0,01  of  an  inch,  and  in  the  best 
to  the  jlo  or  0,001  of  an  inch. 

The  best  time  for  taking  observations  with  the  barometer  for 
the  purpose  of  calculating  heights  is  during  settled  weather  and 
at  mid-day.  Observations  taken  in  the  morning  or  evening  are 
much  more  liable  to  be  erroneous  on  account  of  ascending  and 
descending  currents  of  air  which  take  place  at  these  times. 
Moreover  a  course  of  continued  observations  is  more  likely  to 
lead  to  accurate  results  than  single  observations.  In  the  calcula- 
tion of  very  small  heights  near  the  level  of  the  ocean,  it  is  common 
where  great  accuracy  is  not  required,  to  dispense  with  the  formula 
and  adopt  the  following  rule,  namely,  as  0,1  inch  is  to  the  difference 
in  the  barometric  columns,  so  is  87  feet  to  the  approximate  difference 
of  level  required ;  which  is  to  be  corrected,  if  necessary,  for  the 
difference  from  31*^  of  the  mean  temperature  of  the  air  at  the  two 
stations. 

Thus,  under  a  pressure  of  30  inches  of  mercury  at  the  tem- 
perature of  50^,  0,1  of  an  inch  of  mercury  answers  to  87  feet  of 
atmosphere.  It  will  be  seen  moreover,  that,  as  0,1  of  an  inch 
of  mercury  is  equivalent  to  87  feet  of  air,  0,01  of  an  inch  answers  to 
8,7  feet ;  0,001  of  an  inch  to  0,87  feet ;  and  -^\s  of  an  inch  to  1,74. 
Hence  in  a  good  mountain  barometer,  graduated  to  500dlhs  of  an 
inch,  there  will  be  a  sensible  difference  in  the  pressure  of  the  air 
arising  from  a  change  of  altitude  of  less  than  two  feet,  or  two 
thirds  the  length  of  the  instrument. 

Formula  (iv.)  is  essentially  the  same  with  that  given  by  Laplace 
in  the  10th  book  of  the  Mecanique  Celeste,  but  simplified  after  the 
example  of  Poisson,  and  reduced  to  English  measures.  Ihe 
following  example  will  serve  to  illustrate  every  part  of  this  formula. 

At  the  lower  of  two  stations,  the  mercury  in  the  barometer 
was  observed  to  be  29,4  inches,  and  its  temperature  50°,  that  of 
the  air  being  45° ;  and  at  the  upper  station,  the  height  of  the  ba- 
rometer was  25,19,  its  temperature  46°,  and  that  of  the  air  39°, 
the  latitude  of  the  place  being  30°. 

In  this  case,  we  have 

T  4-  T  4.'5°4-  39° 

T—  t'  =  50°  — 46°  =  4°j-^-t-i_3lo=  — ^        —31°  =  11^ 

Z  lit 


366 


and 


Hydrostatics. 


cos  2  L  =  cos  2  X  30°  =  cos  60°  = 


Whence 


w  —  29,4     log.  .  .  1,46835 
w'  =  25,2003   log.  .  .  1,40141 


11° 

0,00223      . 

669,4     log..  . 
log..  . 
log..  . 

16,42  log..  . 

2,82569 
1,04139 
3,34830 

1st  correction 

1,21538 

1  0,002837  . 

685,82  log..  . 
log..  . 

0,97  log..  , 

2,83621 
3,15168 

2d  correction 

.  1,98789 

E  =  3481280 

686,79  log..  . 
log..  . 

log..  . 

.      2  log.. 

3,4 
690,19  fathom; 

2,83682 
6,54174 

-  =  0,0002 

R 

3d  correction 

,  4,29508 
..  0,00034 

5. 

Laplace's  formula,  applied  to  the  same  example,  gives  688,97 
fathoms,  differing  from  the  above  only  1,22  fathoms;  whereas,  by 
Sir  George  Shuckburgh's  method,  in  which  no  account  is  taken  of 
the  variation  of  gravity,  either  for  difference  of  latitude  or  difference 
of  elevation  in  the  same  latitude,  the  result  is  685,125.  This  cor- 
responds with  the  approximate  height  derived  from  the  first  correc- 
tion in  the  above  example. 

475.  We  have  already  mentioned,  that,  unless  very  particu- 
lar precautions  are  taken,  mercury  is  depressed  in  glass  tubes, 
and  that  this  depression  is  inversely  proportional  to  the  diame- 
ter of  the  tube.  It  is  always  indicated,  moreover,  when  it  takes 
place  by  the  upper  surface  being  convex.     It  is  not  necessary 


Barometer  applied  to  the  Measurement  of  Heights.        367 

to  have  regard  to  this  circumstance  in  the  calculation  of  heights  by 
the  barometer,  where  the  two  observations  are  taken  with  the  same 
instrument,  since  the  ditference  in  the  length  of  the  barometric 
columns  would  be  the  same,  whether  they  were  corrected  or  not.* 
But  in  order  that  observations  by  different  instruments,  liable  to 
different  capillary  effects,  may  be  strictly  compared  with  each  other, 
a  correction  should  be  applied,  which  may  be  readily  done  by 
means  of  the  following  table. 


Interior  diameter  of  the 

Depression 

tube  in  English  inches. 

of  the  Mercury. 

0,6 

0,005 

0,5 

0,007 

0,4 

0,015 

0,35 

0,025 

0.3 

0,036 

0,25 

0,050 

0,2 

0,067 

0,15 

0,092 

0,1 

0,140 

*  Also  in  a  syphon  barometer,  or  one  in  which  the  tube,  instead 
of  entering  a  basin,  turns  up  at  the  bottom  and  continues  of  the 
same  bore,  as  in  figures  230,  232,  since  the  capillary  effect  is  the 
same  in  both  branches,  the  observed  altitude  reckoned  from  the 
surface  in  the  shorter  branch,  would  not  be  affected  by  the  correc- 
tion. 


HYDRODYNAMICS. 


Of  the  Discharge  of  Fluids  through  Apertures  in  the  Bottom  and 
Sides  of  Vessels. 

476.  If  a  fluid  be  made  to  pass  through  a  canal  or  tube  of 
variable  bore,  kept  constantly  full,  and  the  velocity  be  the  same 
in  every  part  of  the  same  section,  since  for  any  given  time  the 
same  quantity  of  fluid  must  pass  through  every  section,  this  quantity 
must  be  equal  to  the  area  of  the  section  multiplied  by  the  veloci- 
ty, a,  a,  being  the  areas  of  two  sections,  and  v,  v',  the  velocities 
at  these  sections,  we  shall  have 

a  V  =.  a'  v', 
and  hence 


that  is,  the  velocities  in  different  sections  are  inversely  as  the  areas  of 
the  sections. 

The  case  here  supposed  is  purely  theoretical,  and  can  never 
occur  in  practice,  since  on  account  of  friction,  the  velocity  is 
always  greatest  at  the  surface  in  a  canal,  and  at  the  axis  in  a 
tube. 

477.  Let  MNOP  represent  a  vessel  filled  with  a  fluid  up  to  Fig.  233. 
GH,  CD  an  aperture,  very  small  compared  with  the  bottom 
MP,  CIKD  the  column  of  fluid  directly  above  the  aperture,  and 
CABD  the  lowest  lamina  or  stratum  of  this  fluid,  immediately 
contiguous  to  the  aperture.  Also  let  v  denote  the  velocity  ac- 
quired by  a  heavy  body  in  falling  freely  through  BD,  the  height 
of  the  stratum,  and  u  the  velocity  which  the  same  stratum  would 
Mech.  47 


370  Hydrodynamics. 

acquire  in  falling  through  the  same  space  by  the  pressure  of  the 
column  CIKD.  If  we  suppose  the  lowest  stratum  ACDB,  to 
fall  as  a  heavy  body  through  the  height  BD,  the  moving  force 
will  be  its  own  weight.  But  if  we  suppose  it  to  be  urged  by  its 
own  weight,  together  with  the  pressure  of  the  incumbent  column 
of  fluid  CIKD,  through  the  same  space,  the  velocity  in  the  former 
case  will  be  to  that  in  the  latter,  as  the  moving  forces  and  the  times 
in  which  they  act,  the  mass  moved  being  the  same  in  both  cases. 
27.  But  the  moving  forces  are  to  each  other,  as  the  heights  BD,  KD, 
and  the  times  in  which  they  act,  the  space  being  the  same,  are 
inversely  as  the  velocities.     Accordingly, 

BD      KD 

V  :  u  :  :  :  • 

V  u 


264 


Whence 


or 


1)2  :  ti^  ::  BD  :  KD, 
V  :  u  :\  s/BD  :  x^KD. 


Now  V  is  the  velocity  which  a  heavy  body  would  actually  ac- 
quire in  falling  through  the  space  BD,  and  as  the  velocities, 
other  things  being  the  same,  are  as  the  square  roots  of  the  spa- 
268.  ces,  u  the  velocity  of  the  issuing  fluid  is  that  which  a  heavy  body 
would  acquire  in  falling  through  KD,  the  height  of  the  fluid  above 
the  orifice.  Therefore,  the  velocity  with  which  a  fluid  is  discharg- 
ed from  the  bottom  of  a  vessel  is  equal  to  that  acquired  by  a  heavy 
body  in  falling  through  a  space  equal  to  the  height  of  the  fluid  above 
the  orifice.  Also  if  a  pipe  A'B'C'D'  be  inserted  horizontally, 
or  inclined  in  any  way  to  the  horizon,  it  may  be  shown,  in 
like  manner,  since  the  pressure  of  fluids  is  equal  in  all  directions, 
that  the  fluid  will  be  discharged  with  the  same  velocity  as  be- 
fore. It  will  accordingly  ascend  to  the  level  of  the  fluid  in  the 
vessel,  all  obstructions  being  removed  ;  and  it  is  found  in  fact, 
under  the  most  favorable  circumstances,  nearly  to  reach  this 
point.  It  follows,  moreover,  from  what  is  above  laid  down,  that 
if  apertures  be  made  at  different  distances  s,  s',  s",  below  the 
surface,  the  velocities  at  these  points,  and  consequently  the 
quantities   of  fluid   discharged   at  these  points,   from    apertures  of 


Discharge  of  Fluids  through  Apertures.  371 

the  same  size,  in  the  same  time,  the  vessel  being  kept  filled  to 
the  same  level,  will  be  as  \/s,  \/s',  \/s".  The  actual  velocity  at 
the  distance  s  below  the   surface  of  the  fluid   in  the  vessel  will  be 


\/2gs,  and  the   quantity  Q  discharged  in  the   time  t,  through  an    gr? 
aperture  whose  area  is  a,  is  as  follows,  namely, 


q  =  at^^gs. 

478.  What  is  above  said  of  the  velocity  of  a  fluid  discharged 
from  jets  or  apertures,  is  true  only  of  the  middle  filament  of 
particles  issuing  through  the  centre  of  the  aperture,  which  are 
supposed  to  experience  no  retardation,  and  which,  in  fact,  suffer 
no  other  retardation  than  what  arises  from  the  resistance  of  the 
air,  and  their  mutual  adhesion  and  attrition  against  each  other. 
But  those  which  issue  near  the  edges  of  the  aperture  suffer  a 
much  greater  resistance,  and  are  accordingly  much  more  retarded. 
Hence  it  follows  that  the  mean  velocity  of  the  whole  column 
of  discharged  fluid  will  be  considerably  less  than  that  indicated  by 
the  above  theory. 

479.  Sir  Isaac  Newton  discovered  a  contraction  in  the  vein  of 
discharged  fluid,  and  found  that  at  a  distance  from  the  orifice  of 
about  a  diameter  of  this  orifice,  the  section  of  the  vein  or  stream 
was  diminished  nearly  in  the  ratio  of  \/2  to  \/l.  Hence  he 
concluded  that  the  velocity  of  the  fluid  after  passing  the  aperture 
was  increased  in  this  proportion,  the  same  quantity  passing  through 
a  narrower  space  in  the  same  time. 

According  to  some  very  accurate  experiments  of  Bossut,  the 
actual  discharge  through  a  hole  made  in  the  side  or  bottom  of  the 
vessel,  is  to  the  theoretical  as  1  to  0,62,  or  nearly  as  8  to  5.  The 
theoretical  discharge  must,  therefore,  be  diniinished  in  this  ratio  lo 
obtain  the  actual  discharge. 

If  the  water  issues,  not  through  an  aperture  in  the  side  or 
bottom  of  the  vessel,  but  through  a  pipe  from  1  to  2  inches  in 
length,  inserted  in  the  aperture,  the  contraction  of  the  vein  is  pre- 
vented, and  the  actual  discharge  becomes  to  the  theoretical,  as  8  to 
10,  or  as  4  to  5.  In  this  way,  therefore,  the  discharge  is  increased 
nearly  in  the  ratio  of  4  to  3. 


372  Hydrodynamics. 

The  theoretical  discharge,  the  discharge  through  an  additional 
tube,  and  that  through  a  simple  perforation  in  the  side,  are  as  the 
numbers  16,  13,  and  10  nearly. 

480.  When  an  upright  cylinder  or  prismatic  vessel  is  suffer- 
ed gradually  to  discharge  itself,  the  velocity  of  the  descending 
surface  of  the  fluid  is  to  the  velocity  at  the  orifice,  as  the  area 
of  the  latter  is  to  that  of  the  former,  and  this  is  a  constant  ratio  ; 
consequently  the  velocity  of  the  descending  surface  varies  as  the 
velocity  at  the  orifice,  or  as  \/s ;  that  is,  the  velocity  of  the 
descending  surface  varies  as  the  square  root  of  the  space  to  be 
described  by  it ;  so  that  this  corresponds  exactly  to  the  case  of 
a  body  projected  perpendicularly  upward ;  whence,  as  the  re- 
270.  tarding  force  is  constant  in  the  instance  just  referred  to,  it  must 
be  constant  also  in  the  case  before  us.  Therefore,  ivhen  a  vessel 
of  (he  above  description  is  suffered  to  discharge  itself,  the  velocity  of 
the  descending  surface  and  that  of  the  discharged  fluid  will  he  uni- 
formly retarded. 

Suppose  a  body,  urged  by  a  constant  force,  as  that  of  gravity, 
to  describe  a  space,  as  1  rod,  for  instance,  in  the  first  second ; 
2G7.  the  spaces  being  as  the  squares  of  the  times,  it  will  describe  4 
rods  in  two  seconds,  9  rods  in  3  seconds,  and  so  on ;  and  the 
spaces  described  in  the  .first  second,  second  second,  &c.,  will 
evidently  be  the  differences  of  these,  namely,  I  —  0,  4  —  J , 
9  —  4,  16  —  9,  &;c.,  that  is,  the  series  of  odd  numbers,  1,  3,  5,  7, 
9,  &.C.  Accordingly,  these  numbers,  taken  in  the  inverse  order, 
represent  the  spaces  described  in  equal  times  by  a  body  uniform- 
ly retarded  ;  they  represent,  moreover,  as  will  be  seen  from  what 
is  above  jiroved,  the  quantities  of  fluid  discharged  in  equal  times 
from  an  aperture  in  the  bottom  of  a  prismatic  vessel.  Hence, 
if  it  were  proposed  to  construct  a  clepsydra,  or  water-clock,  by 
means  of  a  prismatic  or  cylindrical  vessel,  having  an  aperture 
Fig.234,  in  the  bottom,  let  the  height  DB  of  a  vessel  which  would  be 
completely  exhausted  in  a  given  time,  as  12  hours,  be  divided 
from  the  top  downward  into  portions  represented  by  the  numbers 
23,  21,  19,  Uc,  down  to  1,  which  will  require  the  height  DB  to 
be  divided  into  144  equal  parts,  and  these  portions  23,  21,  Stc, 
will  be  the  spaces  through  which  the  upper  surface  will  descend 
in  each  successive  hour  of  the  exhaustion. 


Discharge  of  Fluids  through  Apertures.  373 

481.  If  a?  denote  the  space  through  which  the  upper  surface 
A  descends  in  the  time  t,  the  velocity  of  the  discharged  fluid, 
represented  by  ^2  g  [s  —  x),  will  vary  continually,  but  may  be 
considered  as  constant  during  the  indefinitely  small  time  d  t ;  so 
that  in  this  time  there  will  escape  through  the  orifice,  a  prism 
of  the  fluid  having  the  area  a  of  this  orifice  for  its  base,  and 
\/2  g  {s  —  x)  for  its  altitude.  Thus  the  quantity  discharged  dur- 
ing the  instant  d  i  is  a  d  t  \/2  g  {s  —  x).  But  during  the  same  time 
the  upper  surface  has  descended  through  the  space  d  x,  and  the 
vessel  has  lost  a  prism  or  cylinder  of  the  fluid,  whose  base  is  a,  and 
altitude  d  x,  and  whose  bulk  or  volume  is  a  d  x,  whence 

A  d  X  =  a  d  i  y'a  g{s  —  x), 

and 

,                 A  dx  .   . 

d  t  = -.  (i.) 

As  the  area  a  will  be  given  in  functions  of  x,  by  the  form  of  the 
vessel,  the  second  member  of  this  equation  may  be  considered  as 
containing  only  the  variable  quantity  x,  and  it  will  be  easy  in  most 
cases,  by  simply  integrating,  to  discover  the  successive  depressions 
of  the  surface,  and  the  discharges  of  the  fluid,  from  any  vessel  of  a 
known  form. 

482.  Let  the  vessel,  for  example,  be  an  upright  prism  or  cylin- 
der ;  a  in  this  case  will  be  constant,  and  we  shall  have 

a  /*     d  X  2  a        , ,     ^ 

t  =  =-  .  /  -— := =^  \/s  —  X  -f-  o. 

a  v2g    t/    vs  —X  ^V^g 

Now  when  die  time  t  is  0,  the  depression  of  the  upper  surface 
A  is  0  also  ;  thus  we  have  at  the  same  time  x  =:  0,  and  t  z=  0  ', 
this  condition  determines  the  constant  quantity  C  to  be 

2  a        -- 


a  v2 


and  gives  for  the  time  of  depressing  the  upper  surface  through  the 
space  X,  as  follows, 


Geom. 
290. 


374  Hydrodynamics. 

To  find  the  time  of  completely  emptying  the  vessel,  we  have 
only  to  make  a?  =  s,  in  which  case  the  preceding  expression  will 
become 

a  v/2g  a  ^2g  ^  ^  \  ^ 

But  from  what  is  above  shown,  we  have  Qor  a  s  =  a  t  \/2gs, 
from  which  we  obtain 


t  = 


a  ^2gs 


By  comparing  this  result  with  the  preceding,  it  will  be  seen  that 
when  a.  vessel  is  svffered  to  exhaust  itself,  the  time  employed  is 
just  double  that  required  to  discharge  the  same  quantity  when  the 
vessel  is  kept  full.  The  same  conclusion  might  indeed  be  drawn 
from  articles  266,  270. 

483.  Let  the  vessel  be  any  solid  generated  by  the  revolution  of 
a  curve.  The  axis  being  vertical,  a  will  be  the  area  of  a  circle 
which  has  for  its  radius  the  ordinate  y  of  the  generating  curve,  that 
is,  if  n  =  3,14159  he,  a  =z  n  y^.  Substituting  this  value  for  a  in 
the  equation  (i.)  of  article  481,  we  have 


\/s  — X 


V 

In  any  particular  examples,  it  will  be  necessary  lo  put  for  y 
its  value  deduced,  in  terms  of  x,  from  the  equation  of  the  generat- 
ing curve. 

484.  Let  ABCD  be  the  vertical  side  of  a  vessel,  EFGH  a 
Fig  235.  rectangular  notch  in  it,  and  let  IL  i  I  be  a  rectangular  parallelo- 
gram whose    breadth  1  i  is  infinitely  small  compared   with   EG. 
The  velocity  with  which  the  fluid  would  escape  at  GH,  is  to  the 
velocity  with  which  it  would  escape  from  IL  i  I,  as  \/EG  to  \/EI, 


Discharge  of  Fluids  through  Apertures.  375 

and    the    quantities  of  fluid    discharged  in  a   given  time    through    268. 
indefinitely   small   parallelograms  at  these  depths  are  in   the  same 
ratio.     But  the   parabolic  curve  jEZH  being   drawn,   having  EG 
for  its  axis,  we  have 

EG  :  EI  ::  GH  :  Ik] 

and  consequently,  Trig.nc. 

X/EG  :  ^/Ei  ::   GH  :  IK', 

whence  the  quantities  discharged  through  indefinitely  small  par- 
allelograms at  the  depths  EG,  EI,  are  to  each  other  as  the  ordi- 
nates  GH,  IK,  and  the  sum  of  all  the  quantities  discharged 
through  all  the  parallelograms  of  which  the  rectangle  EFGH  is 
composed,  is  to  the  sum  of  all  the  quantities  discharged  through 
as  many  parallelograms  at  the  depth  EG,  as  the  sum  of  all  the 
elements  IKk  i  of  the  parabola,  to  the  sum  of  all  the  correspond- 
ing elements  IL  I  i  of  the  rectangle  ;  that  is,  as  the  area  of  the 
parabola  EKHG  to  the  area  of  the  rectangle  EFGH ;  in  other 
words,  the  quantity  running  through  the  notch  EFGH  is  to  the 
quantity  running  through  an  equal  horizontal  area  at  the  depth 
EG,  as  EKHG  to  EGHF,  that  is,  as  2  to  3.  Therefore  the  Cal.  94. 
mean  velocity  of  the  fluid  in  the  notch  is  equal  to  two  thirds  of  that 
at  the  greatest  depth  GH 

485.  If  a  small  aperture  be  made  in   the  side  of  a  vessel  keptFig.236. 
filled  to  the  same  height,  the  fluid  will  spout  out  horizontally  with 
the  velocity  acquired    by  a  heavy  body  in  falling  freely  through  the 
height  of  the  fluid  above  the  aperture,  and  this   velocity  combin-    477. 
ed    with    the   perpendicular   velocity   arising    from   the    action    of 
gravity,  will  cause   each  particle,  and  consequently  the  whole  jet 
to  describe  a   parabola.     Now  the  velocity  with   which   the  fluid    303. 
is  expelled  from  any  aperture,  as  G,  is  such  as  would,  if  uniform- 
ly  preserved,  carry  a   particle   through  a   space   equal  to  2  BG    ^66. 
in  the  time  of  its  natural  descent  through  BG  ;  accordingly,  if  the 
direction  of    the    aperture    be    horizontal,    the    action   of    gravity 
being    at    right   angles   to    it   will  cause    the   particle  to  descend 
through  the   height   GD  in   the  same  time   that  would    be  required 
in  case  of  a  natural  descent  through  GD,  if  no  other  force  were 


376  Hydrodynamics. 

267.  exerted  upon  the  particle.  Hence  the  squares  of  the  times  being 
as  the  spaces,  or  the  limes  simply  as  the  square  roots  of  the  spaces, 
\/^G  is  to  \/'GD  as  the  time  employed  in  describing  BG  to  the 
time  required  to  reach  the  horizontal  plane  DF.  But  in  the  time 
employed  in  describing  BG,  the  particle  would  be  carried  uniform- 
ly and  horizontally  by  the  velocity  thus  acquired,  through  a  space 
equal  to  2BG;  therefore,  to  find  the  amplitude  or  horizontal 
range  DE  of  the  jet,  we  have  the  proportion 

2.r'  \/BG  :  x/GD  :  :  2  BG  :  DE  =  ^^^^^ 

=  2  \/BGGD  =  2  GH. 

As  the  same  reasoning  may  be  used  with  respect  to  any  other 
point  in  BD,  ij  upon  the  height  of  the  fluid  BD  as  a  diameter  we 
describe  a  semicircle  BKD,  the  horizontal  distance  to  which  the  fluid 
will  spout  from  any  point  will  be  twice  the  ordinate  of  the  circle 
drawn  through  this  point,  the  distance  being  measured  on  the  plane  of 
the  bottom  of  the  vessel. 

486.  It  will  hence  be  perceived,  that  if  apertures  be  made  at 
equal  distances  G,  L,  from  the  top  and  bottom  of  the  vessel, 
the  horizontal  distances  DE  to  which  the  fluid  will  spout  from 
these  apertures  will  be  equal ;  and  that  the  point  I,  bisecting 
the  altitude,  is  that  from  which  the  fluid  will  spout  to  the  greatest 
distance,  this  distance  DF  being  equal  to  twice  the  radius  of  the 
semicircle  or  to  the  altitude  BD  of  the  fluid. 

487.  If  the  fluid  issue  obliquely  instead  of  horizontally,  the 
curve  described  will  still  be  parabolic,  and  the  horizontal  range, 
&,c.  of  the  jet  may  be  calculated  as  in  the  case  of  other  projectiles. 

Fig.237.  Let  the  aperture  C  be   inclined,  for  example,  upward   at    different 

angles.      CB  will  be   equal  to  5,  the  space  through  which  a   body 

must  fall  to  acquire  the  velocity  of  projection,  and  equal  to  the  dis- 

477.    tance   CF,    CF',   of  the  foci  of  the   several   parabolas,  traced   by 

particles   issuing  with   different   angles  of  elevation.     Hence  BE 

Trig.172.  is  the  directrix  to  these  parabolas,  and  the  circle  described  from 
the  centre  C,  and  with  the  radius  BC,  will  pass  through  the  sev- 
eral foci  -F,  jP^  he.  Let  CE,  for  instance,  be  the  direction  of 
the  jet,  and  draw  CF  making  the  angle  ECF  equal  to  BCE-,  let 


Motion  of  Gases.  377 

fall  the  perpendicular  FH,  and  take  HG  equal  to  HC,  the  dis- 
tance CG  will  be  the  horizontal  range  of  the  jet.     But 
CG  =  2CH=2  CF  X  cos  FCH  =  2  CB  x  sin  2  ECF. 

Therefore,  when  the  angle  of  elevation  is  45°,  the  focus  of  the  par- 
abola falls  on  the  horizontal  line  at  F^,  and  the  range  CK  is  then 
the  greatest  possible,  being  double  the  altitude  CB. 


Of  the  Motion  of  Gases. 

488.  To  determine  with  what  velocity  the  air  or  any  other  gas 
will  rush  into  a  void  space,  when  urged  by  its  own  weight,  we  pro- 
ceed according  to  a  method  analogous  to  that  by  which  the  motion 
of  liquids  is  determined.  When  the  moving  force  and  the  mass  or 
matter  to  be  moved  vary  in  the  same  proportion,   the   velocity  will 

continue  the  same,  since  v  =  — .  28. 

7n 

Thus,  if  there  be  similar  vessels  of  air  and  water,  extending 
to  the  top  of  the  atmosphere,  on  the  supposition  of  a  uniform 
density  throughout,  they  will  be  discharged  through  equal  and 
similar  apertures  with  the  same  velocity ;  for  in  whatever  pro- 
portion the  quantity  of  matter  moving  through  the  aperture  be 
varied  by  a  change  of  density,  the  pressure  which  forces  it  out 
acting  in  circumstances  perfectly  similar  will  vary  in  the  same 
proportion.  Hence  it  follows  that  the  air  rushes  into  a  void  with 
the  velocity  which  a  heavy  body  woidd  acquire  by  falling  from  the 
top  of  the  atmosphere,  this  fluid  being  supposed  to  be  of  a  uniform 
density  throughout. 

The  height  of  a  uniformly  dense  or  homogeneous  atmosphere 
being  27807  feet,  according  to  article  4G7,  and  g  =  32,2,  we 
shall  have  for  the  velocity  in  question 


V  =  x^2g  h  =  x/^  X  32,2  X  27807  =  1338.  277. 

489.  But  as  the  space  into  which  the  air  rushes  becomes  more 
and  more   filled   with  air,  the  velocity  must  be  diminished   contin- 
ually.    Indeed  whatever  be  the  density  of  this  rarer  air,  its  elas- 
ticity, varying  with    its   density,   will    balance   a   proportional   part 
Mech.  48 


378  Hydrodynamics. 

of  the  pressure  of  the  atmosphere,  and  it  is  the  excess  of  this 
pressure  only  which  constitutes  the  moving  force,  the  matter  to 
be  moved  being  the  same  as  before.  Let  d  be  the  natural  den- 
sity of  the  atmosphere,  and  A  the  density  of  that  which  opposes 
itself  to  the  motion  in  question.  Let  p  be  the  pressure  of  the 
atmosphere,  or  the  force  which  impels  it  into  a  void,  and  C7  the 
force  with  which  this  rarer  air  would  rush  into  a  void  ;  from  the 
proportion 

p  A 

we  shall  have,  for  the  moving  force   sought,  p  —  - — .    Again,  let 

V  be  the  velocity  of  air  rushing  into  a  void  under  the  pressure  jo, 
and  u  the  velocity  of  air  under  the  sam.e  pressure  rushing  into 
rarefied  air  of  the  density  A.  Since  the  pressures  are  as  the  heights 
producing  them,  the  fluid  being  supposed  of  a  uniform  density 
throughout,  we  shall  have 


whence  u  =  v  X      (   1 ,110  allowance  being  made  for  the 

inertia  of  the  rarer  air,  which  being  displaced  must  oppose  a  certain 
resistance. 

490.  Let  it  be  proposed  to  determine  the  time  t  in  seconds  in 
which  the  air  will  flow  into  a  given  exhausted  vessel,  until  the  air 
shall  have  acquired  in  the  vessel  a  certain  density  a. 

Suppose  h  the  height  due  to  the  velocity  v,  b  the  bulk  or 
capacity  of  the  vessel,  and  a  the  area  of  the  aperture,  the  meas- 
ure in  each  case  being  in  feet.  Since  the  quantity  of  air  neces- 
sary to  fill  the  vessel  will  depend  upon  the  size  of  the  vessel,  and 
also  upon  the  density  of  the  air,  b  A  will  represent  this  quantity, 
the  differential  of  which  is  &  rf  A.  The  velocity  of  influx  at  the 
first  instant  is  v  =  \/2gh ',  and  when  the  air  in  the  vessel  has 
acquired  the  density  A,  that  is,  at  the  end  of  the  time  t,  the 
velocity  is 


V^gh    X   Jl— ^  0Y^2gh  X  - 


—  A 


Motion  of  Gases.  379 

Hence  the  rate  of  influx,  which  may  be  measured  by  the  infinitely 
small  quantity  of  air  passing  the  aperture  during  the  instant  d  t  with 
this  velocity,  will  be  denoted  by  • 


J 


D  —  A 


2g  h  X   ■ X  J)  a  d  t  =o  d  t  ^2  g  d  h  {d  —  A). 


D 


Putting  these  two  values  of  the  rate  of  influx  equal  to  each  other, 

we  have 


a  d  t  \/2  g  D  fi  (p  —  A)  =b  d  A, 

and 

,  b  dA 

d  t  = X :. 

o  \/'Z  g  T»  h         ^/d  —  A 

Hence,  by  integrating,  we  obtain 
b 


t  = X  \/i^  —  A  -f-  C. 

To  find  die  constant  C,  it  will  be  observed,  that  when  t  =z  0, 
A  =  0,  and  \/i)  —  A  ;=:  y^D.  We  have,  therefore,  for  the  cor- 
rected integral 

t  —  — X  ( Vd  —  V'^  —  A  ) . 

a  ^/^  gv  h         \  / 

491.  When  d  =  A,  the   motion  ceases,  and  die  value  of  t,  or 
the  time  of  completely  filling  the  vessel,  becomes 

b  v/d                       b                        b  , 

or ,  or  -,  nearly. 


o  v/y  ^  D  A  a  s/}  g  h  4  (J  ^//t 

Suppose,  for  example,  die  capacity  of  the  vessel  to  be  8  cubic 
feet,  or  nearly  a  wine  hogshead,  and  that  the  aperture  by  which 
air  of  the  ordinary  density,  or  I,  enters,  is  an  inch  square,  or  j\^ 
of  a  foot.  In  this  case  4  y'A  :=  4  ^^27807  =  668,  nearly  ;  and 
hence 


380  Hydrodynamics. 

If  the  aperture  be  only  y|u  of  a  square  inch,  or  the  side  y'^,  the 
time  of  completely  filling  the  vessel  will  be  172''  nearly,  or  a 
little  less  than  3'.  • 

If  the  experiment  be  made  with  an  aperture  cut  in  a  thin  plate, 
479.     vve  shall  find  the  time  greater  nearly  in  the  ratio  of  62  or  G3  to  100, 
as  we  have  already  remarked  with  respect  to  water  flowing  through 
small  orifices. 

492.  We  can  find,  in  like  manner,  the  time  necessary  for 
bringing  the  air  in  the  vessel  to  any  particular  density,  as  |  of 
that  of  air  in  its  ordinary  state.  For  the  only  variable  part  of 
the  integral,  above  found,  is  -v^"  —  ^f  which  in  this  case  becomes 
/^l  —  f  =:  J,  and  gives  \/i>  —  \/d  —  a  =  ^  ;  hence,  if  the 
aperture  were  a  square,  each  side  being  j\  of  an  inch,  the  time 
sought  would  be  J  172'',  or  SG",  nearly. 

493.  If  the  air  in  the  vessel  be  compressed  by  a  weight 
acting  on   the   movable  cover  ^D,  the   velocity  of  the   expelled 

Fig.23S.  air  may  be  determined  thus.  Let  the  additional  pressure  be 
denoted  by  q,  and  the  density  thence  resulting  by  d'  ;  we  shall 
then  have 

J}  :  p  -]-  q  :  :  n   :  T>', 

and 

P  •  P  ~\~  9  —  P  •  •  D   •  d'  —  D, 

which  gives 

d' D 

9=PX   -^  . 

Now,  since  the  pressure  which  expels  the  air  is  the  difference 
between  the  force  which  compresses  the  air  in  the  vessel  and  that 
which  compresses  the  internal  air,  die  expelling  force  is  q ;  whence, 
the  forces  being  as  the  quantities  of  motion, 

d'  —  D 

p  :  2>  X   :  :  m  V  :  n  u, 

m,  n,  being  the  masses  expelled,  v  the  velocity  with  which  air 
rushes  into  a  void,  and  u  the  velocity  required.  But  the  masses 
or  number  of   particles   which  issue  through  the   same  orifice  in 


Motion  of  Gases.  381 

an   instant,   are   as   the  densities  and   velocities  conjointly ;.  hence 


and  consequently, 


p  :  p   X    :  :   D  t;^  :  d'  u^, 

which  gives 

Jd'  —  D 
ri. 
d' 

Moreover,  from  the  proportion 


we  obtain 


and 


Whence 


P  '  p  +  q 


ij/  —  ^  (P  +  9) 


I  »  (P  +  9) 

D'  D   =  ^^ — ' — i^  D 

P 

P   (P   +  g)  —  Dp     _   »? 
P  ~    P' 


Pg 


»  (l?  +  3)  P   +  ? 


Substituting  this  value  of -, —  m    the    above    expression   for  m, 

we  have  the  following  simple  and  convenient  formula,  namely. 


u  =  V      !_?_.. 
\P  +  9 

494.  We  have  taken  no  notice  of  the  effect  of  the  air's  elasti- 
city upon  the  velocity  of  influx  into  a  void.  Let  ABCD  be  a 
vessel  containing  air  of  any  density  d.  This  air  is  in  a  state  of 
compression,  and  if  the  compressing  force  be  removed  it  will  ex- 
pand, and  its  elasiicity  will  diminish  with  the  density.  Now  its 
elasticity,   in    whatever   state,   is   measured     by   the    force   which 


382  Hydrodynamics. 

keeps  it  in  that  state  ;  and  the  force  which  keeps  common  air  at 
its  ordinary  density  is  the  pressure  of  the  atmosphere,  and  equiv- 
alent to  that  of  a  colunm  of  mercury  30  inches  in  height.  If, 
therefore,  we  suppose  this  air,  instead  of  being  confined  by  the 
top  of  the  vessel,  to  be  pressed  down  by  a  movable  piston  car- 
rying a  cylinder  of  mercury  of  the  same  base  and  30  inches 
high,  its  elasticity  will  balance  this  pressure  just  as  it  does  the 
pressure  of  the  atmosphere ;  and  since  from  its  fluidity  the  pres- 
sure received  on  any  one  part  is  propagated  through  every  part 
and  in  every  direction,  it  will  press  on  any  small  portion  of  the 
vessel  by  its  elasticity,  as  when  loaded  with  this  column.  Hence, 
if  this  small  portion  of  the  vessel  be  removed  leaving  an  opening 
into  the  void,  the  air  will  begin  to  flow  out  with  the  same  velo- 
city as  it  would  flow  out  when  pressed  by  its  own  weight  only, 
or  with  the  velocity  acquired  by  falling  from  the  top  of  a  ho- 
mogeneous atmosphere,  or  1338  feet  per  second.  But  as  soon 
as  a  portion  of  air  had  passed  through  the  orifice,  the  density  of 
that  remaining  in  the  vessel  being  reduced,  its  elasticity,  and 
consequently  the  expelling  force,  is  diminished.  But  the  matter 
to  be  moved  is  diminished  in  the  same  proportion  as  the  density, 
the  capacity  of  the  vessel  remaining  unchanged  ;  therefore,  since 
the  density  and  elasticity  follow  the  same  law,  the  mass  moved 
will  vary  as  the  moving  force,  and  the  velocity  will  continue  the 
same  from  the  beginning  to  the  end  of  the  efflux. 

495.  The  velocity  with  which  the  air  issues  out  of  a  vessel 
under  the  circumstances  above  supposed,  being  constant,  we  can 
readily  compare  the  velocity  given  by  the  theory  with  that 
Fig.  239.  found  by  experiment.  Let  .^  be  a  cask  of  known  capacity  in 
the  top  of  which  is  an  aperture  a  of  a  known  area.  The  tube 
TB,  recurved  at  B,  is  soldered  or  screwed  into  the  top  of  the 
cask.  The  aperture  a  is  stopped  while  water  is  poured  into  the 
tube  T  till  it  is  full,  at  which  time  a  quantity  of  water  will  have 
passed  out  at  B  condensing  the  air  in  the  cask  till  its  spring  is 
equal  to  the  weight  of  the  water  in  the  tube.  At  this  time  a 
cock  placed  over  the  tube  T,  sufiiciently  large  to  supply  water 
as  fast  as  it  can  descend  into  the  vessel  ^i,  is  to  be  opened  to 
keep  the  tube  constantly  filled.  For  this  purpose  one  person 
must  constantly  tend  it,  while  another  opens  the  aperture  a, 
which  needs  only  to  be  closed  with  the   finger,  the  seconds   being 


Motion  of  Gases.  383 

counted  from  the  moment  the  finger  is  removed  till  the  water 
flies  out  at  a.  Hence,  knowing  the  capacity  of  the  vessel  and 
the  area  of  the  aperture,  we  obtain  the  velocity.  If  the  tube  TB 
should  be  continued  nearly  to  the  bottom  of  j1,  while  A  was  filling 
with  water,  the  length  of  the  compressing  column  would  be 
gradually  diminished,  and  consequently  the  pressure  constantly 
changing.  To  avoid  any  irregularity  from  this  cause  the  open 
end  of  the  tube  is  placed  as  near  the  top  of  the  cask  as  is  consis- 
tent with  a  free  passage  for  the  water. 

The  vessel  was  made  to  contain  IS""-  6°^-  of  water,  from  which 
its  capacity  is  found  to  be  425,083  cubic  inches.  The  area  of 
the  aperture  a  through  which  the  water  is  discharged  was  0,0046 
inches. 

(1.)  The  altitude  of  T  above  the  cask  being  30  inches,  the 
time  of  expelling  the  air  was  found  by  several  trials  to  be  33". 

(2.)  The  altitude  of  T  being  6  feet,  the  time  of  expelling  the 
air  was  21,3'^ 

In  the  first  experiment,  425,088,  the  capacity  of  the  cask,  being 
divided  by  0,0046,  the  area  of  the  aperture,  gives  92410,4  inches 
for    the    length   of    the    stream    continued    during   33''.     Hence 

92410  4 

— r '-^  =  233,3    feet,  the  velocity  per  second. 

From  the  second  experiment  we  deduce  by  a  similar  process, 
361,6  for  the  velocity  per  second  ;  and  to  show  the  correspondence 
of  this  with  the  first,  we  use  the  proportion 

V'Sl  :  V^  :  :  233,3  :  361,8, 
differing  from  the  experimental  result  one  fifth  of  a  foot. 

496.  To  compare  the  velocity  thus  found  by  experiment  with 
that  assigned  by  theory,  we  use  the  proportion 

Ve  :  \/34  :  :  361,6  :  860,5, 

the  velocity  with  which  the  atmosphere  would  begin  to  enter  a 
void.  Taking  the  result  before  found,  namely,  1338,  and  multi- 
plying it  by  0,63,  agreeably  to  what  is  laid   down  in  article  479, 


384  Hydrodynamics. 

we  shall  have  842,94,  differing  from  the  experimental  resuik  about 
A  part. 

497.  Let  it  be  proposed  to  find  the  quantity  of  air  expelled 
Fig.  238. into  an  infinite  void  from  the  aperture  C  of  the   vessel  ABCD 
during    any  time  i,  and  the    density  of  the  remaining  air   at  the 
end  of  that  time. 

The  bulk  expelled  during  the  instant  d  t  will  he  a  d  t  \/  2  g  h, 
the  velocity  \/  2  g  h  being  constant,  and  consequently  the  quantity 
will  he  a  d'  d  t  \/  2  g  h.  The  quantity  at  the  beginning  of  the 
efflux  is  6  D,  i  being  as  before  the  bulk  of  the  vessel ;  and  when  the 
air  has  acquired  the  density  d',  the  quantity  in  the  vessel  is  b  a', 
and  the  quantity  expelled  is  Z»  d  —  b  d'  ;  consequently,  the  quan- 
tity discharged  during  the  instant  d  t  must  be  the  differential  of 
J  D  —  b  a',  that  is,  —  b  d  d'.     Hence  we  have  the  equation 

a  T>'  d  t  A^2  g  h  =■  —  b  d  J)', 

and 

,     b  d  v>'         b  do' 

~~         a  v'   x/2gh  ~  a   \/2  g  h  d'  ' 

the  integral  of  which  is 

a   x^2  g  h 

h.    log.    denoting   the  hyperbolic  or   Naperian    logarithm    of    d''. 
When  ^  =  0,  d'  is  equal  to  d  ;  whence 


therefore. 


nearly. 


c  =  , h.  log.  D  ; 

a  ^2gh  ^       ' 


,  b  ,     .        D  6        ,     ,        D 


It  is  obvious  that  no  finite  time  will  be   sufficient  for  the   ves- 
sel to  empty  itself;  for,  as  d'  must  in  this  case  be  equal   to  zero, 

—  will  be  infinite,  and  its  logarithm  will  also  be  infinite  ;  so  that 

t  will  be  infinite. 


Motion  of  the  Air.  385 

498.  It  is  by  a  train  of  reasoning  precisely  similar,  that  we 
ascertain  the  quantity  of  condensed  air  which  will  make  its  escape 
from  a  vessel  into  the  atmosphere  in  a  given  time.  Let  A  be  the 
density  of  the  condensed  air,  and  h'  the  height  of  a  homogeneous 
atmosphere  corresponding  to  it;  also  let  d  be  the  density  of  the 
atmosphere.  The  air,  having  for  its  density  A,  will  obviously,  when 
mixing  with  the  atmosphere,  have  the  same  velocity  as  though  it 
were  rushing  into  a  vacuum  with  the  density  A  —  d.  Now  the 
height  of  a  homogeneous  fluid  corresponding  to  this  density  is  found 
by  the  proportion 

A  :  A  —  D  :  ;  A'   :  /i'  ^  —  ^, 


From  the  equation  v  =  \/2  g  s,  it  will  be  seen  that  the  velo- 
cities acquired  by  falling  from  different  heights  are  proportional 
to  the  square  roots  of  these  heights.  If,  therefore,  v  be  taken  to 
represent  the  velocity  of  common  air  rushing  into  a  vacuum,  and 
h  the  height  of   the  corresponding  homogeneous  atmosphere,  we 

shall  have,  u  being  the  velocity  belonging  to  the  height  h'  — - — , 
A D  Q         2 2  '*'     ■^  —  o 


h'  — ^ —  :  :  V    :  u^  z=  v^  —  . 

A  /t  A 


whence 


^  =  ^  Jr- 


A—  D 


Moreover  the  proportion 

V  :  A  :  :  h  :  h', 


gives 


therefore 


h'  _  A 
h  D 


J  A      A  —  D_  |A  —  D 

D    '  A  X  D        ' 


Let   A'  be  the   density  of  the   condensed   air  after  the  time  t,  b 
being,  as  before,  the   bulk  of   the  vessel,  and  u  a  section  of  the 
Mcch.  49 


386  Hydrodynamics. 

aperture,  we  shall  have 

— - —  =  —b  d  A', 

and 

—  bdA'        _  —  6  n/d    ^,  d  A 

d  t  ^=^ =^  — 7. —  X 


•A'  D  O  V  A'   y/A'  —  D 

aV  A'- 7= 

Let  \/A'  —  D  =  X,  then  a'  —  d  =  x%  and  A'  =  x^  +  d, 
from  which  we  have 

d  A'  z=z  2  X  d  X, 

and  hence 

d  A'  2xdx  2  dx 


A'  ^A'  D  (x2   +  d)  Z  x2  _|_  ^p2 

2  rf  a;  1 

Cdi.  120.  Now  the  integral  of  -o—. — j^  is  equal  to  -r=  muhiplied    by  au 

arc  whose  tangent  is  -^,  radius  being  1 . 
Whence 


—  2  6  </Ti  1  1  .    \/^'  —  »    , 

t  =  - — —  X  —r^  X  an  arc  whose  tang,  is  j= \-  c. 

a  V  v^D  -v/d 

When 

i  =  0,  A'  is  equal  to  A, 

and 


26  , 

c  =  —  X  an  arc  whose  tans;. 


JA— D 
D 


Let  the  former  arc  be  represented   by  a  and  the  latter  by  a' ;  we 

shall  have 

2  6 
t  =  —  (a'  —  a). 

When   the   lime  is  required  in   which  the   density  of  the  air 
contained  in  the  vessel   shall  be  reduced  to  that  of  the  external 
atmosphere  j  as  A'  =  d,  in  this  case,  it  follows  that  a  =:  0, 
and 

_  2  6  g^ 


Motion  of  Gases.  387 

To  illustrate  this  by  an  example,  let  it  be  required  to  find 
the  time  in  which  air  of  double  the  atmospheric  density,  confined 
in  a  cubical  vessel,  each  of  whose  sides  is  12  feet,  will  expand  into 
the  atmosphere  through  an  opening  of  one  tenth  of  an  inch  in  dia- 
meter, so  as  to  be  reduced  to  the  common  density.  The  above 
formula  gives,  by  substitution, 

2  X  123 
^  =  0,007854  X  1338  ^  ^"  ^'"^  ^^°'^  ^^"S-  >s  1, 

2  X  123  X  100 
=  0,7854  X   13"38  ^  ^"  ''^  '^^°'"  *""§•  's  1, 


2  X  123  X  100 
1338 


258^'  =  4M8' 


499.  Even  although  the  density  of  the  confined  air  were 
infinitely  greater  than  that  of  the  atmosphere,  the  time  in  which  it 
would  be  reduced  would  be  a  finite  quantity.  For  A  being  infinite- 
ly greater  than  d,  the 


is  also  infinite,  and  corresponds  to  an  arc  of  90°.  Hence  in  this 
case  we  have 

t  =—  X  2  X   0,7854. 

cr  V 

The  capacity  of  the  vessel  and  the  area  of  the  aperture  being  the 
same  as  in  the  last  example,  we  should  have  for  the  time  in  which 
this  infinitely  condensed  air  would  be  reduced  to  the  same  density 
with  the  atmosphere 

'  =  0.007^4^1338  X  ^  X  0,7854  =  5C1"  =  8'  36". 


388  Hydrodynamics. 


Of  the  Resistance  of  Fluids  to  Bodies  moving  in  them. 

500.  The  force  with  which  solid  bodies  moving  in  fluids,  as 
water,  air,  &c.,  are  impeded  and  retarded,  is  usually  termed  the 
resistance  of  fluids ;  and  as  all  our  machines  move  either  in  water 
or  in  air,  or  both,  it  becomes  a  matter  of  importance  in  the 
theory  of  mechanics  to  inquire  into  the  nature  of  this  kind  of 
force. 

We  know  by  experience  that  force  must  be  applied  to  a  body 
in  order  that  it  may  move  through  a  fluid,  such  as  air  or  water ; 
and  that  a  body  projected  with  any  velocity  is  gradually  retard- 
ed in  its  motion,  and  generally  brought  to  rest.  We  also  know 
that  a  fluid  in  motion  will  hurry  a  solid  body  along  with  it,  and 
that  force  is  necessary  to  maintain  the  body  in  its  place.  And 
as  our  knowledge  of  nature  teaches  us  that  the  mutual  actions 
of  bodies  are  in  every  case  equal  and  opposite,  and  that  the  ob- 
served change  of  motion  is  the  only  indication  and  measure  of 
the  changing  force,  we  infer  that  the  force  which  is  necessary 
to  keep  a  body  immovable  in  a  stream  of  water,  flowing 
with  a  certain  velocity,  is  the  same  with  that  which  is  requir- 
ed to  move  this  body  with  an  equal  velocity  through  stagnant 
water. 

A  body  in  motion  appears  to  be  resisted  by  a  stagnant  fluid, 
because  it  is  a  law  of  mechanical  nature  that  force  must  be 
employed  in  order  to  put  any  body  in  motion.  Now,  the  body 
cannot  move  forward  without  putting  the  conliguous  fluid  in 
motion,  and  force  is  to  be  used  to  produce  this  motion. 
In  like  manner,  a  quiescent  body  is  impelled  by  a  stream  of 
fluid,  because  the  motion  of  the  contiguous  fluid  is  diminished  by 
this  solid  obstacle ;  the  resistance,  therefore,  or  impulse,  differs 
in  no  respect  from  the  ordinary  communication  of  motion  among 
solid  bodies,  at  least  in  its  nature  ;  although  it  may  be  far  more 
difficult  to  reduce  the  various  circumstances  attending  it  to  accu- 
rate computation,  or  to  obtain  all  the  requisite  data  on  which  to 
found  the  calculation 


Resistance  of  Fluids.  389 

501.  The  resistance  which  a  body  suffers  from  the  fluid 
medium  through  which  it  is  impelled  depends  on  the  velocity, 
form,  and  magnitude  of  the  body,  and  on  the  inertia  and  tenacity 
of  the  fluid.  For  fluids  resist  the  motion  of  bodies  through  them, 
(1.)  by  the  inertia  of  their  particles;  (2.)  by  their  tenacity,  that  is, 
the  adhesion  of  those  particles ;  (3.)  by  the  friction  of  the  body 
against  the  particles  of  the  fluid.  In  perfect  fluids  the  two  last 
causes  of  resistance  are  very  inconsiderable,  and  therefore  are 
not  taken  into  the  account ;  but  the  first  is  always  very  con- 
siderable, and  obtains  equally  in  the  most  perfect  as  in  the  most 
imperfect  fluids.  And  that  the  resistance  varies  with  the  velo- 
city, shape,  and  magnitude  of  the  moving  body  is  sufficiently 
obvious. 

We  must  carefully  distinguish  between  resistance  and  retard- 
ation ;  resistance  is  the  quantity  of  motion,  retardation  the  quan- 
tity of  velocity^  which  is  lost ;  therefore,  the  retardations  are  as 
the  resistances  applied  to  the  quantities  of  matter ;  and  in  the  same 
body  the  resistance  and  retardation  are  proportional. 

502.  To  determine  the  force  of  fluids  in  motion,  or  the  resist- 
ance of  fluids  against  bodies  moving  in  them. 

(1.)  In  fluids  uniformly  tenacious,  the  resistance  is  as  the 
velocity  with  which  the  body  moves.  For,  since  the  cohesion  of 
the  particles  of  the  fluid  is  always  the  same  for  the  same  space, 
whatever  be  the  velocity,  the  resistance  from  this  cohesion  will 
be  as  the  space  described  in  a  given  time  j  that  is,  as  the 
velocity. 

(2.)  In  a  fluid  whose  particles  move  freely  without  disturb- 
ing each  others'  motions,  and  which  flows  in  behind  as  fast  as  a 
plane  body  moves  forward,  so  that  the  pressure  on  every  part  of  the 
body  is  the  same  as  if  the  body  were  at  rest,  the  resistance  will  be 
as  the  density  of  the  fluid. 

(3.)  On  the  same  hypothesis  the  resistance  will  be  as  the 
square  of  the  velocity.  For  the  resistance  must  vary  as  the  number 
of  particles  which  strike  the  plane  in  a  given  time,  multiplied  into 
the  force  of  each  against  the  plane ;  but  both  the  number  and  the 
force  is  as  the  velocity,  and  consequently  the  resistance  is  as  the 
square  of  the  velocity. 


390  Hydrodynamics. 

This  proof  supposes  that  after  the  body  strikes  a  particle,  the 
action  of  that  particle  entirely  ceases ;  whereas  the  particles, 
after  they  are  struck,  must  necessarily  diverge,  and  act  upon  the 
particles  behind  them ;  thus  causing  some  difference  between 
theory  and  experiment.  This  hypothesis,  however,  on  account  of 
its  simplicity,  is  generally  retained,  and  corrected  afterwards  by 
deductions  from  actual  experiments. 

This  ratio  of  the  square  of  the  velocity  may  be  otherwise 
derived,  thus. 

It  is  evident,  that  the  resistance  to  a  plane,  moving  perpen- 
dicularly through  an  infinite  fluid,  at  rest,  is  equal  to  the  pressure 
or  force  of  the  fluid  on  the  plane  at  rest,  the  fluid  moving 
with  the  same  velocity,  and  in  the  contrary  direction  to  that  of 
the  plane  in  the  former  case.  But  the  force  of  the  fluid  in  mo- 
tion must  be  equal  to  the  weight  or  pressure  which  generates 
that  motion  ;  and  which,  it  is  known,  is  equal  to  the  weight  or 
pressure  of  a  column  of  the  fluid,  whose  base  is  equal  to  the 
plane,  and  its  altitude  equal  to  the  height  through  which  a  body 
must  fall  by  the  force  of  gravity,  to  acquire  the  velocity  of  the 
fluid  ;  and  that  altitude  is,  for  the  sake  of  brevity,  called  the  alti- 
tude due  to  the  velocity.  So  that  if  a  denote  the  surface  of  the 
plane,  v  the  velocity,    and  s  the  specific  gravity  of  the   fluid  ;  then 

the  altitude  due  to  the  velocity  v  being  ^ — .  the  whole  resistance 

or  moving  force  m,  will  be 

w^  a  s  v^ 


ff  X  s  X  ^r-  — 


2g  2g    ' 

_g-  being  32,2  feet.     And  hence,  other  things  being  the  same,  the 
resistance  is  as  the  square  of  the  velocity. 

(4.)  If  the  direction  of  the  motion,  instead  of  being  perpendicu- 
lar to  the  plane,  as  above  supposed,  be  inclined  to  it  at  any  angle, 
then  the  resistance  to  the  plane  in  the  direction  of  the  motion,  as 
assigned  above,  will  be  diminished  in  the  triplicate  ratio  of  radius 
to  the  sine  of  the  angle  of  inclination,  or  in  the  ratio  of  1  to  i  ^, 
where  i  is  the  sine  of  the  inclination. 


Resistance  of  Fluids.  391 

For  AB  being  the  direction  of  the  plane,  and  bd  that  of  the  ^'o-  ^^' 
motion,  abd  the  angle  whose  sine  is  i;  the  number  of  particles 
or  quantity  of  the  fluid  which  strikes  the  plane  will  be  diminish- 
ed in  the  ratio  of  I  to  i;  and  the  force  of  each  particle  will 
likewise  be  diminished  in  the  same  ratio  ;  so  that  on  both  these 
accounts  the  resistance  will  be  diminished  in  the  ratio  of  1  to  i  ^  ; 
that  is,  in  the  duplicate  ratio  of  radius  to  the  sine  of  abd.  But 
further,  it  must  be  considered  that  this  whole  resistance  is  exerted 
in  the  direction  be  perpendicular  to  the  plane  ;  and  any  force  in 
a  direction  be  is  to  its  effect  in  a  direction  ae,  parallel  to  bd,  as  ae 
to  be,  or  as  1  to  i.  Consequently,  on  all  these  accounts,  the  resist- 
ance in  the  direction  of  the  motion  is  diminished  in  the  ratio  of 
1  to  P.  And  if  this  be  compared  with  the  result  of  the  preced- 
ing step,  we  shall  have  for  the  whole  resistance,  or  the  moving 
force,  on  the  plane, 

a  s  v^  i'^ 


m  = 


2^ 


(5.)  If  w  represent  the  weight  of  the  body  whose  plane  sur- 
face a  is  resisted  by  the  absolute  force  m,  then  the  retarding 
force 

J. m   a  s  v^  i^ 

J        w  2  g  w  ' 

(6.)  And  if  the  body  be  a  cylinder  whose  surface  or  end  is 
a,  and  diameter  d,  or  radius  r,  moving  in  the  direction  of  its 
axis;  then,  because  i  =  1,  and  a  =  tt  R^  =  i  w  d^,  where 
TT  =  3,141593,  the  resisting  force  m  will  be 

TT  S   D^  1)2  TT  8   r2  y2 


&g  ^g 


71  S   J)^  V^  TT  S    r2   U^ 


and  the  retarding  force 

/■= . 

8  g  w  2  g  w 

(7.)  This  is  the  value  of  the  resistance  when  the  end  of  the 
cylinder  is  a  plane  perpendicular  to  its  axis,  or  to  the  direction 
of  the  motion.  But  were  its  face  a  conical  surface,  or  an  elliptic 
section,  or  any  other  figure  every  where  equally  inclined  to  the 
axis,  the  sine  of  the  inclination  bein^  i ;  then  the  number  of  pani- 
cles of  the  fluid  striking  the  surface  being  still  the  same,  but  the 
force  of  each,  opposed  to  the  direction  of  the  motion,  diminished 


392  Hydrodynamics. 

in  the  duplicate  ratio  of  the  radius  to  the  sine  of  the  inclination^ 
the  resisting  force  m  would  be 

TT  s  D^  u-  i^        71  s  n^  v^  i^ 


Sg  2g 

But  if  the  body  were  terminated  by  an  end  or  surface  of  any 
other  form,  as  a  spherical  one,  where  every  part  of  it  has  a  different 
inclination  to  the  axis ;  then  a  further  investigation  becomes 
necessary. 

503.  To  determine  the  resistance  of  a  fluid  to  any  body  moving 
in  it,  having  a  curved  end  as  a  sphere,  a  cylinder  with  a  hemispheri- 
cal end,  &fc. 

Fig. 241,  (!•)  Let  BEAD  be  a  section  through  the  axis  ca  of  the  solid, 
moving  in  the  direction  of  that  axis.  To  any  point  of  the  curve 
draw  the  tangent  eg,  meeting  the  axis  produced  in  g  ;  also  draw 
the  perpendicular  ordinates  ef,  e  /,  indefinitely  near  to  each 
other ;  and  draw  a  e  parallel  to  cg. 

Putting  OF  =z  0?,  EF  =  y,  be  =  z,  i  =  sine  of  the  angle  g, 
radius  being  1  ;  then  2  n  y  is  the  circumference  whose  radius  is  ef, 
or  the  circumference  described  by  the  point  e,  in  revolving  about 
the  axis  ca  ;  and  2  n  y  X  e  e,  or  2  tt  3/  c?  z,  is  the  differential  of 
the  surface,  or  it  is  the  surface  described  by  e  e,  in  its  revolution 
about  ca  ;  hence 

X  2  n  y  d  z,  or X  y  d  z 


2g     —••:'"'-'-'        g 

is  the  resistance  on  that  ring,  or  the  differential  of  the  resistance 
to  the  body,  whatever  the  figure  of  it  may  be  ;  the  integral  of 
which  will  be  the  resistance  required. 

(2.)  In  the   case  of  a  spherical  shape ;  putting  the  radius  ca 
or  c  B  =  R,  we  have 

V/     o  o\  •  EF  CF  X 

^  '  EG  CE  R 

and  yrfzorEF  X  Ee  =  CE  X  a  e  ^=.  Vid  x; 

therefore  the  general  differential 

7r  s  w^      .  _      , 

.  t^  y  d  z 

g  ^ 


Resistance  of  Fluids.  393 

becomes 

7t  S  V"^       X^  ,  TT  s  w2  _ 

.  -X  .  B.  a  X  =  5-  .  x"*  a  x: 

g  R-i  g   R'^ 

the  integral  of  which,  or  j ^  x^,  is  the  resistance  to  the  spheri- 
cal surface  generated   by  be.      And  when  a:  or  cf  is  =  b  or  ca, 

O        o 
IT  S   ?)"    R. 

it  becomes  — j for  the  resistance  on  the  whole  hemisphere  ; 

which  is  also  equal  to  —^3 5  where  d  =  2  r,  the  diameter. 

(3.)  But  the  perpendicular  resistance  to  the  circle  of  the 
same   diameter  d  or  bd,  by  section  6  of  the   preceding  problem,  is 

— ;=r ;  which  beina;  double  the  former,   shows  that   the  resist- 

8  g     '  => 

ance  to  the  sphere  is  just  equal  to  half  the  direct  resistance  to  a  great 

circle  of  it,  or  to  a  cylinder  of  the  same  diameter. 

(4.)  Since  i  ti  d^  is  the  magnitude  of  the  globe  ;  if  s'  denote 
its  density  or  specific  gravity,  its  weight  w  will  be  =:  |  w  d^  s',  and 
therefore  the  retarding  force 

~      m        71  s  u^  D^         6  3  s  u^ 


10  16  g       *  7r  s'  D^  8   ^  s'  D  ' 

which  is  also  equal  to 


2gs 


Hence 


and 


3s  1 


4  s'  D        s 


_  s 
s 


which  is  the  space  that  would  be  described  by  the  globe  while 
its  whole  motion  is  generated  or  destroyed  by  a  constant  force 
equal  to  the  resistance,  if  no  other  force  acted  on  the  globe  to 
continue  its  motion.  And  if  the  density  of  the  iluid  were  equal 
to  that  of  the  globe,  the  resisting  force  sufficient  would  be  act- 
ing constantly  on  the  globe  without  any  other  force,  to  generate 
or  destroy  the  motion  while  describing  the  space  ^  d,  or  f  of  its 
diameter. 

Mcch.  50 


394  Hydrodynamics. 

(5.)  Hence  tlie  greatest  velocity  that  a  globe  acquires  in  de- 
scending through  a  fluid,  by  means  of  its  relative  weiglit  in  the 
fluid,  will  be  found  by  putting  the  resisting  force  equal  to  that 
vtreight.  For,  after  the  velocity  has  arrived  at  such  a  degree  that 
the  resisting  force  is  equal  to  the  weight  that  urges  it,  it  will  increase 
no  longer,  and  the  globe  will  afterwards  continue  to  descend  with 
that  velocity  uniformly.  Now,  s'  and  s  being  the  specific  gravities 
respectively  of  the  globe  and  fluid,  s'  —  s  will  be  the  relative 
gravity  of  the  globe  in  the  fluid,  and  therefore 

W  =  1  71  D^  (s'  —  s) 
is  the  weight  by  which  it  is  urged  ;  also 


m  ■=. 


71  s  i;2  D^ 


is  the  resistance  ;  consequently 


2  r.2 

=  i  71  d3  (s^  _  s), 


16^ 

when    the    velocity   becomes    uniform  ;    from    which   equation    is 
found 

»=J(2^.Jr>.^), 

for  the  above  uniform  motion  or  greatest  velocity. 

By  comparing  this  with  the   general  equations,  v  =  \/g  s,  it 
will  be  seen   that  the   greatest  velocity  is  that   acquired    by   the 

accelerating   force    ,    in  describing  the  space   |  d,  or  that 

acquired  by  gravity  in  describing  freely  the  space 


If  s'  =:  2  s,  or  the  specific  gravity  of  the  globe  be  double  that 

of  the  fluid,  '■ =  1  =:  the  natural  force  of  gravity ;    and  the 

globe  will  attain  its  greatest  velocity  in  describing  f  d  or  f  of  its 
diameter.  It  is  further  evident,  thnt  if  the  globe  be  very  small, 
it  will  soon  attain  its  greatest  velocity,  whatever  its  density 
may  be. 


Resistance  of  Fluids.  '  395 

If  a  leaden  ball,  for  example,  one  inch  in  diameter  descend 
in  water,  and  in  air  of  the  usual  density  at  the  earth's  surface,  the 
specific  gravities  of  these  bodies  being  11,3,   1,   0,00122,  respec- 

s'  "^^  s 

lively,  since  f  of  an  inch  is  -^\,  or  0,11,  of  a  foot, becomes 

10,3  in  the  case  of  water,  and 

11,3  —  0,00122        „^^         , 
0,00122         =  ^'^  "^^^'y' 

in  the  case  of  air,  we  shall  have 


y  =  V2  X  32,2  X  0,11  X  10,3  =  8,54 

nearly,  for  the  greatest  velocity  the  ball  can  acquire  per  second  in 
water  :   and 


V2  X  32,2  X  0,11  X  926  =  256 

nearly,  for  the  greatest  velocity  in  air. 

But  if  the  globe  were  only  one  hundredth  of  an  inch  in  diam- 
eter, the  greatest  velocities  would  be  only  j-\  of  the  above  re- 
sults, or  0,85  of  a  foot  in  water,  and  25,G  in  air  ;  and  if  the  ball 
were  still  further  diminished,  the  greatest  velocity  would  be  di- 
minished also,  in  the  subduplicate  ratio  of  the  diameter  of  the  ball. 
This  is  well  illustrated  in  the  fall  of  rain.  The  different  sized 
drops  descend  with  different  degrees  of  rapidity,  but  all  so  gently  as 
to  cause  no  injury.  Were  this  fluid  so  constituted  as  to  allow  the 
drops  to  form  in  larger  masses,  or  were  the  air  much  less  dense, 
tender  vegetables  would  suffer  by  the  shock,  as  they  sometimes  do 
in  fact  by  the  more  rapid  descent  of  hail. 

504.  It  appears  from  the  third  step  of  the  preceding  article, 
that  the  resistance  to  the  motion  of  a  cylinder  moving  in  the 
direction  of  its  axis  is  double  that  of  a  globe  of  the  same  diame- 
ter ;  and  in  experiments  with  bodies  that  move  slowly,  this  will 
nearly  hold  true  in  water,  but  still  more  nearly  in  air ;  because 
its  particles  move  more  freely  than  those  of  water,  and  less  dis- 
turb each  others'  motions ;  but  when  the  motion  is  more  rapid, 
considerable  aberrations  will  occur,  both  from  the  mutual  dis- 
turbance of  the  particles,  and  from  the  fluid  not  flowing  in  so  fast 
behind  as  the  body  moves  forward  ;  in  the  air  also,  a  new  cause 
of  deviation  will  arise,  from  the  condensation  of  the  fluid  before 


396  Hydrodynamics. 

the  body.  Sir  Isaac  Newton  supposes,  that  in  a  continuous  non- 
elastic  fluid,  infinitely  compressed,  the  resistances  of' a  sphere 
and  cylinder  of  equal  diameters  arc  equal ;  but  this  appears  to 
be  an  error  in  theory  as  well  as  in  fact.  When  the  motion  is 
slow  in  water,  the  fluid  may  be  conceived  to  be  nearly  of  the 
nature  which  Newton  supposes ;  yet  the  resistances  are  almost 
as  coincident  with  theory  as  when  the  motion  is  in  air ;  thus  M. 
Borda  found  the  resistance  of  a  sphere  moving  in  water  to  be  to 
that  of  its  greatest  circle  as  1  to  2,508,  and  in  air  the  resistances 
were  as  1  to  2,45.  The  experiments  of  Dr.  Hutton  in  air  give 
the  resistances  at  a  mean  as  1  to  2^. 

The  reason  that  experiment  gives  the  ratio  of  the  resistances 
greater  than  that  of  2  to  1  seems  to  be  this ;  in  theory  it  is  sup- 
posed that  the  action  of  every  particle  of  the  fluid  ceases  the  instant 
it  makes  its  impact  on  the  solid ;  but  this  is  not  actually  the  case,  as 
we  have  before  observed  ;  and  since  the  particles,  after  impact  on 
the  sphere,  slide  along  the  curved  surface,  and  hence  escape  with 
more  facility  than  along  the  face  of  the  cylinder,  the  error  will 
be  greater  in  the  cylinder ;  that  is,  the  greater  resistance  will  ex- 
ceed the  theory  more  than  the  less.  It  is  also  to  be  observed, 
that  the  difference  between  the  resistances  of  the  globe  and  cyl- 
inder in  water  is  greater  than  in  air ;  and  this  is  directly  contrary 
to  what  might  be  inferred  from  Newton's  reasoning,  which  sup- 
poses them  equal  in  a  continuous  fluid,  but  in  the  ratio  of  1  to  2 
in  a  rare  fluid. 

505.   To  determine  the  relations  of  velocity,  space,  and  time,  of  a 
ball  moving  in  a  fluid  in  which  it  is  projected  with  a  given  velocity. 

Let  u  =  the  velocity  of  projection,  s  the  space  described 
in   any  time  t,  and  v  the  velocity  acquired.      Now,  by  step   4,   arti- 

cle  503,  the  accelerating  force  f  =■  — -, —  ;  where  s'  is  the  den- 

8  ^  S    D 

sity  of  the  ball,  s  that  of  the  fluid,  and  d  the  diameter  of  the 
ball.     Therefore  the  general  equation  v  d  v  =z  g  fd  s  becomes 

7  —  3  s  «2 

V  d  V  =— d  s; 

8  S'  D 

and  hence 

dv  _  —  3  s     7     _  7 

V     ~    8  s'  -D  ~~  ' 


Resistance  of  Fluids.  397 

8  s'd 


o  _ 

putting  c  for  q-j—'     The  correct  integral  of  this  is  log.  u  —  log.  v 


or  log.    -  =  cs.     Or  putting  e  =  2,718231828,  the  number  whose 


u 

V 

u 


hyp.  log.  is  1,  then   -  =  e",  and  the  velocity 


u 


506.  The  velocity  v  at  any  time  being  the  e""^  part  of  the  first 
velocity,  the  velocity  lost  in  any  time  will  be  the  1  — e~'^^  part,  or 

the    — - —  part  of  the  first  velocity. 

(1.)  If  a  globe,  for  example,  be  projected  with  any  velocity  in 
a  medium  of  the  same  density  with  itself,  and  it  describe  a  space 
equal  to  3  d  or  3  of  its  diameters ;  then  5  =  3  d,  and 


3s 

3 

~  8d' 

therefore  c  s  := 

9 

and  the  velocity 

lost  is 

e"  —  1 

2,08 

gC3 

~  3,08' 

or  nearly  |  of  the  projectile  velocity. 

(2.)  If  an  iron  ball  of  two  inches  diameter  were  projected  with 
a  velocity  of  1200  feet  per  second  ;  to  find  the  velocity  lost  after 
moving  through  any  space,   as  500  feet  of  air ;  we  should  have 

D  =:  -r^  =  1,  M  =  1200,  s  —  500,   s'  =  71,  s  =  0,0012 ; 

and  therefore 

_  Sss  _  3  .  0,0012  .  500  _  3  .  12  .  500  .  3  .  6  _     81 

^  *  ~  8  s"  i  —        8  .  7i  .  1       ~"        8  .  22  .  10000     ~    440  ' 

and 

1200 

V  =     _8_i^  =  998  feet  per  second  j 

having  lost  202  feet,  or  nearly  1  of  its  first  velocity. 


S80. 


398  Hydrodynamics. 

(3.)  If  the  earth  revolved  about  the  sun,  in  a  medium  as  dense 
as  the  atmosphere  near  the  earth's  surface ;  and  it  were  required 
to  find  the  quantity  of  motion  lost  in  a  year ;  since  the  earth's 
mean  density  is  about  4i,  and  its  distance  from  the  sun  12000  of 
its  diameters,  we  have 

24000  X  3,141G  =  75398  diameters  =  s, 

and 

_  3  .  75398  .  12  .  2  _  ^  ^_.^ 
""  '  -      8  .  10000  .  9      -  ^'^"^-^^  ' 

gCS      J 

hence    — - —  =  -i|f|  parts  are  lost  of  the  first  motion  in  the 

space  of  a  year,  and  only  the  ^^-^^j  part  remains. 
(4.)  To  find  the  time  t ;  we  have 

J     d  s  d  s  e'^^  d  s 

V  u  u      ' 


Now,  to  find  the  integral  of  this,  put  z  =  e"^  ;  then  is  c  s  =  log.  z, 
and 


consequently 


and  hence 


,  6?  2  ,  d  z 

c  d  s  =1  — ,  or    d  s  ■=.  — 

2  C  Z 


J  ^        e'^^  d  s         z  d  z        d  z 

a  t  or  =:  z=  — 

u  u  u  c 


_    z     _  e"^ 
u  c         «  c    ' 


But  as  t  and  s  vanish  together,  and  when  s  =  0,  the  quantity 


—  IS  equal  to   — 
u  c        *  u  c 


therefore 


—  !!ljiii  —  i L  —  1  /"i 1\ 

u  c  cv        c  u         c  \v         u } 

the  time  soudit :  where  c  =  ^   ,    ■>  and  «  =  -—  the  velocity. 

^     '  o  s  D  e''° 


Air-Pump.  399 


On  the  Theory  of  the  Mr-Pump,  and  Pumps  for  raising  Water. 

607.  The  Air-Pump  is  a  machine  fitted  to  exhaust  the  air 
from  a  proper  vessel,  and  thus  to  produce  what  is  called  a  va- 
cuum ;  it  is  one  of  the  most  useful  of  philosophical  experiments. 
By  means  of  it  the  chief  propositions  relative  to  the  weight  and 
elasticity  of  the  air  are  proved  experimentally,  in  a  simple  and 
satisfactory  manner. 

EFGH  represents  a  square  table  of  wood,  *4,  .^  two  strong  Fig. 244, 
barrels  or  tubes  of  brass,  firmly  retained  in  their  position  by  the 
cross-piece  TT,  which  is  pressed  on  them  by  screws  O,  O,  fixed  on 
the  tops  of  the  brass  pillars  JV,  N.  These  barrels  communicate 
with  a  cavity  in  the  lower  part  D  of  the  table.  At  the  bottom 
within  each  barrel  is  fixed  a  valve,  opening  upwards ;  and  in 
each  barrel  a  piston  works,  having  a  valve  likewise  opening  up- 
wards. The  pistons  are  moved  by  a  cog-wheel  in  the  piece  TT, 
turned  by  the  handle  B,  of  which  wheel  the  teeth  catch  in  the 
racks  of  the  pistons  C,  C.  PQ  is  a  circular  brass  plate,  having 
near  its  centre  the  orifice  ^  of  a  concealed  pipe  that  communi- 
cates with  the  cavity  at  J) ;  at  F"  is  a  screw  that  closes  the  orifice 
of  another  pipe,  and  which  is  turned  for  the  purpose  of  admitting 
the  external  air  when  required.  LM  is  a  glass  vessel,  from 
which  the  air  is  to  be  exhausted,  and  which  has  obtained  the  name 
of  receiver,  because  it  receives  or  holds  the  subjects  on  which 
the  experiments  are  to  be  made.  This  receiver  is  placed  on  the 
plate  PQ,  and  is  accurately  fitted  to  it  by  grinding,  or  by  means 
of  moistened  or  oiled  leather. 

When  the  handle  B  is  turned,  one  of  the  pistons  is  raised  and 
the  other  depressed ;  consequently  a  void  space  is  left  between 
the  raised  piston  and  the  lower  valve  in  the  corresponding  barrel ; 
the  air  contained  in  the  receiver  LM  communicating  with  the  bar- 
rel by  the  orifice  K  immediately  raises  the  lower  valve  by  its 
spring,  and  expands  into  the  void  space ;  and  thus  a  part  of  the 
air  in  the  receiver  is  extracted.  The  handle  then,  being  turned 
the  contrary  way,  raises  the  other  piston,  and  performs  the  same 
operation  in  the  barrel  containing  it ;  while  in  the  mean  lime  the 


400  Hydrodynamics. 

first  mentioned  piston  being  depressed,  the  air  by  its  spring 
closes  the  lower  valve,  and  raising  the  valve  in  the  piston  makes 
its  escape.  The  motion  of  the  handle  being  again  reversed,  the 
first  barrel  again  exhausts,  while  the  second  discharges  the  air 
in  its  turn  ;  and  thus  during  the  whole  time  the  pump  is  worked, 
one  barrel  exhausts  the  air  ft"om  the  receiver,  while  the  other 
discharges  it  through  the  valve  in  its  piston.  Hence  it  is  evi- 
dent, that  the  air  can  never  be  entirely  exhausted ;  for  it  is  the 
elasticity  of  the  air  in  the  receiver  that  raises  the  valve,  and 
forces  it  into  the  barrel  *;  and  each  operation  can  only  take  away 
a  certain  part  of  the  remaining  air,  which  is  in  proportion  to  the 
quantity  before  the  stroke,  as  the  capacity  of  the  barrel  to  the 
sum  of  the  capacities  of  the  barrel,  receiver,  and  communicafing 
pipe. 

508.  Now  if  we  suppose  no  vapor  from  moisture,  &;c.,  to 
rise  in  the  receiver,  the  degree  of  exhaustion  after*  any  number 
of  strokes  of  the  piston  may  be  determined  by  knowing  the  re- 
spective capacities  of  the  barrel  and  of  the  receiver,  including  the 
pipe  of  communication,  he.  For,  as  we  have  seen  above,  that 
every  stroke  diminishes  the  density  in  a  constant  proportion, 
namely,  as  much  as  the  whole  capacity  exceeds  that  of  the  cyhn- 
der  or  barrel ;  the  exhaustion  will  go  on  in  a  geometrical  progression, 
the  ratio  of  which  is  the  same  as  that  which  the  sum  of  the  capaci- 
ties of  the  receiver  and  barrel  bears  to  that  of  the  receiver ;  and 
this  ratio  of  exhaustion  will  continue  until  the  elasticity  of  the  inclu- 
ded air  is  so  far  diminished  by  its  rarefaction  as  to  render  it  too 
feeble  to  push  up  the  valve  of  the  piston. 

Let  then  the  capacity  of  the  barrel,  receiver,  and  pipe  of 
communication  together  be   expressed  hy  b  -\-  r,   and  that  of  the 

*  Various  contrivances  have  been  adopted  to  facilitate  the  mo- 
tion of  the  valve,  and  thus  to  allow  the  air,  when  in  a  state  of  great 
rarefaction,  to  pass  from  the  receiver  into  the  barrels.  Smeaton 
had  recourse  to  a  valve  presenting  a  broad  surface,  and  supported 
on  tliin  bars.  Others  have  proposed  to  raise  the  valve  mechanically 
by  connecting  it  with  the  piston  in  such  a  manner  that  the  piston 
shall  exert  its  action  at  the  moment  it  reaches  the  top  of  the  barrel. 
A  method  suggested  itself  to  Dr.   Prince  much  more   perfect  and 


Air-Pump.  401 

barrel  alone  by  h,  and  let  1   represent  the  primitive  density  of  the 
air  in  the  pump  ;  we  shall  have 

h  -\-  r  :  r   :  :  1   :   j—- —  z=i  the  density  after  1  stroke  of  the  piston. 


h  -\-  r  :  r 


r  ?'2 

1 — ; —  :  n — ; — xs  =  density  after  2  strokes, 
0  +  r      (6  +  ^) 


h  -{-  r  '.  r  :  '.  ~ — ; —  :  — — ■ — r^  =.  density  after  three  strokes ; 

T 

and  the  «th  power  of  the  ratio  ,         , 

or  -pi —  =:  D,  the  density  after  n  strokes. 

(6  +  r)"  '  ^ 


From  which  we  may  easily  find  the  density  after  any  number 
of  strokes  of  the  piston  necessary  to  rarefy  the  air  a  number  of 
times,  or  to  give  it  a  certain  density  d,  the  primitive  density 
being   1.     For    the    above   equation,  expressed   logarithmically,  is 


or 


n 


X  (log.  r  —  log.  {b  +  r))  =  log.  d  j 


more  simple.  It  dispenses  with  the  valve  entirely  by  extending  the 
barrel  downward  so  as  to  admit  of  the  piston's  descending  below  the 
opening  which  communicates  with  the  receiver,  and  thus  allowing  a 
free  introduction  of  air  into  the  barrel.  The  air  in  this  case  is  ex- 
pelled through  a  valve  at  the  top  of  the  barrel,  opening  upward. 
There  will  still  be  a  limit  however  to  the  exhaustion ;  for  the  air 
cannot  be  forced  through  the  valve  at  the  top  of  the  barrel,  unless 
its  elasticity,  and  consequently  its  density,  produced  by  the  motion 
of  tlie  piston,  exceeds  the  density  of  the  external  air.  To  remove 
this  difficulty  Dr.  Prince  enclosed  the  valve  in  question  in  a  vessel 
furnished  with  a  small  exhausting  apparatus,  by  which  the  valve 
was  relieved  from  the  pressure  of  the  atmosphere.  This  appendage 
is  not  necessary,  except  for  purposes  of  very  accurate  exhaustion. 
Mech.  51 


402  Hydrodynamics. 

consequently 

loff.  D 

n  =  ^ 


log.  r  —  log.  (6  +  r) ' 

in  which  expression  d  will  be  a  fraction.  If  the  number  of  times 
which  the  air  is  rarefied  be  expressed  by  n,  an  integer,  the  loga- 
rithmic equation  will  be 

log.  N 

log.  (b  -\-  r)  —-  log.  r' 

A  further  reduction  of  the  same  theorem  will  furnish  us  with 
the  proportion  between  the  capacities  of  the  receiver  and  barrel, 
when  the  air  is  rarefied  to  the  density  d'  by  a  definite  number  of 
strokes  n  of  the  piston.     For  since  • 

if  we  take  the  nth  root  of  both  members  of  the  equation  we  shall 
have 


6  _|_  ,.  —  v^d'. 

Thus,  if  d'  be  equal  to  jj^\^^,  and  die  number  of  strokes  n  =  1 1. 
we  shall  find 


log.  d' 
11  ~ 


^og-l; 


so  that  r  :  6  -[-  r  :  :  1   :  3,  and  6  :  r  : :  2  :   1, 


Pumps  for  Raising  Water. 

509.  The  term  pump  is  generally  applied  to  a  machine  for 
raising  water  by  means  of  the  pressure  of  the  atmosphere.  Of 
pumps  there  are  a  great  many  different  sorts;  we  shall  speak 
only  of  those  in  most  common  use,  and  shall  give  merely  such 
a  general  description  of  their  construction  as  will  enable  the 
student  to  understand  the  principles  on  which  their  operation 
depends. 


Water-Pumps.  403 

The  pumps  most  generally  used  are,  the  sucking  immp,  the 
lifting  pump,  and  the  forcing  pump ;  these  have  some  parts  in 
common,  and  particularly  the  pistons  and  valves  j  they  will  there- 
fore be  treated  in  a  connected  way. 

The  piston  is  a  body  ^j5  CD  of  circular  base,  which  may  be  Fig.  245. 
moved  through  the  interior  part  of  the  tube  or  body  of  the  pump, 
filling  it  exactly  as  it  moves  along.  The  sucker  or  valve  E  is 
movable  about  a  joint  in  such  a  manner  as  eitlier  to  permit  or  to 
prevent  the  passage  of  the  water,  according  as  it  presses  upwards 
or  downwards.  In  figures  245,  246,  there  are  likewise  valves  in 
the  pistons.  FGHK  is  another  tube  joined  to  the  body  of  the 
pump,  and  is  generally  called  the  j)^p^  o^  sucking  pipe  ;  its  lower 
extremity  is  immersed  in  the  water,  of  which  we  suppose  RS  to  be 
the  horizontal  surface. 

510.  The  sucking  pump  is  represented  in  figure  245.  In  this 
pump,  if  we  suppose  a  power  P  applied  to  the  handle  of  the  piston 
so  as  to  raise  it  from  /  to  C,  the  air  contained  in  the  space 
DVKHGFC  tends  by  its  spring  to  occupy  the  space  that  the  pis- 
ton leaves  void ;  it  therefore  forces  up  the  valve  E,  and  enters  into 
the  body  of  the  pump,  its  elasticity  diminishing  in  proportion  as  it 
fills  a  greater  space.  Hence  it  will  exert  on  the  surface  GH  of 
the  water  a  less  effort  than  is  made  by  the  exterior  air  in  its 
natural  state  upon  the  surrounding  parts  of  the  same  surface  RG, 
US  ;  and  the  excess  of  pressure  on  the  part  of  the  exterior  air 
will  cause  the  water  to  rise  in  the  upper  pipe  GK  to  a  certain 
height  HN,  such  that  the  weight  of  the  column  GN,  together  with 
the  spring  of  the  superincumbent  air  shall  just  be  a  counterpoise 
to  the  pressure  of  the  exterior  air.  At  this  time  the  valve  E 
closes  of  itself;  and  if  the  piston  be  lowered,  the  air  contained 
between  the  piston  and  the  base  It^  of  the  body  of  the  pump, 
having  its  density  augmented  as  the  piston  is  lowered,  will  at 
length  have  its  density,  and  consequently  its  elasticity,  greater 
than  that  of  the  exterior  air;  this  difference  of  elasticity  will 
constitute  a  force  sufficient  to  push  the  valve  L  of  the  piston 
upwards,  and  some  air  will  escape  till  the  exterior  and  interior 
air  are  reduced  to  the  same  density.  The  valve  L  then  falls 
again;  and  if  we  again  elevate  the  piston,  the  water  will  be 
raised  higher  in  FGHK,  for  the  same  reason  as  before.     Thus, 


404  Hydrodynamics. 

after  a  certain  number  of  strokes  of  the  piston,  the  water  will 
reach  the  body  of  the  pump  ;  where  being  once  entered,  it  will 
be  forced  at  each  stroke  of  the  piston  through  the  spout  X  ;  for 
the  water  above  the  piston  will  then  press  upon  the  valve  and 
keep  it  shut  while  the  piston  is  rising;  so  that  a  cylinder  of  water 
whose  height  is  equal  to  the  stroke  OT  of  the  piston  (or  the 
vertical  distance  through  which  it  passes)  will  be  raised  by  each 
upward  motion  and  forced  through  the  aperture  X,  provided  it  is 
of  an  adequate  magnitude. 

511.  The  lifting  pump  is  represented  in  figure  246.  Its  man- 
ner of  operation  is  this;  the  piston  PCD  is  here  placed  below 
the  horizontal  surface  RS  of  the  water,  and  when  it  is  made  to 
descend,  it  produces  a  vacuum  between  the  valve  E  (which  is 
pushed  down  by  the  exterior  air)  and  the  base  CD  of  the  piston. 
The  weight  of  the  water,  together  with  that  of  the  exterior  air 
about  R  and  S,  presses  up  the  valve  L,  and  the  water  passes  into 
the  body  of  the  pump;  and  when  the  water  ceases  to  enter,  the 
weight  of  the  valve  L  closes  it.  Then,  if  the  piston  be  raised,  it 
raises  all  the  water  above  it,  forces  up  the  valve  E,  and  intro- 
duces the  water  into  the  part  IVYX.  When  the  piston  is  raised  to 
its  highest  position,  the  valve  E  is  made  to  close  by  the  super- 
incumbent water,  and  retains  the  fluid  there  until,  by  a  fresh 
stroke  of  the  piston,  more  water  is  forced  upwards  through  the 
valve  -E;  that  which  was  before  in  the  upper  part  of  the  pump 
being  expelled  through  a  proper  orifice  or  spout  in  the  neigh- 
bourhood of  X,  in  order  to  make  way  for  a  new  supply.  By 
continuing  the  operation,  water  is  delivered  at  every  stroke  of 
the  piston. 

612.  The  forcing  pump  unites  in  some  measure  the  properties 
of  the  other  two.  The  piston  ^5  CD,  which  here  has  no  valve, 
being  elevated,  rarefies  the  air  in  the  space  DGHVOC,  and  the 
water  rises  towards  K ;  the  subsequent  descent  of  the  piston 
forces  some  of  the  air  in  this  space  through  tlie  valve  L ;  the 
next  ascent  of  the  piston  closes  the  valvfe  L,  and  raises  the  water 
in  GK;  and  so  on  till  the  water  passes  through  the  valve  E  and 
enters  the  space  DIOC.  Then  the  piston  being  pushed  down, 
closes  the  valve  JC,  and  some  of  the  condensed  air  is  forced 
through  the  valve  L.     A  further  stroke  raises   more   water   into 


Water-Pumps.  405 

the  space  DOIC,  and  expels  more  air  through  L.  At  length  the 
water  reaches  Z,,  and  the  subsequent  strokes  raise  it  into  the 
tube  MO  m  n,  whence  it  is  carried  off  by  a  spout,  as  in  the 
other  pumps.  Or,  if  this  pump  be  closed  at  m  n,  excepting  a 
narrow  pipe  PS,  when  the  water  is  raised  by  the  process 
just  described  to  o  r,  above  the  bottom  »S  of  the  tube,  the  elastic 
force  of  the  compressed  air  in  the  space  n  r  o  m  will  compel  the 
water  to  issue  from  the  aperture  P  in  a  continued  stream  or  jet, 
thus  forming  a  fire  engine  or  artificial  fountain. 

513.  Let  us  now  enquire  into  the  fundamental  properties  of 
these  machines.  By  means  of  the  lifting  pump,  water  may  be 
elevated  to  any  height  we  please,  provided  we  employ  a  suffi- 
cient force.  But  the  estimate  of  this  force  requires  various  con- 
siderations. We  must  have  regard  to  the  dimensions  of  the  pis- 
ton, the  barrel  of  the  pump,  the  height  to  which  the  water  is  to 
be  raised,  and  the  velocity  with  which  the  water  is  to  be  moved, 
beside  the  effects  of  friction,  &;c.  At  present,  however,  we  shall 
not  examine  these  particulars  in  all  their  extent ;  but  shall 
confine  ourselves  to  one  of  them.  Now  it  is  certain  that  the 
power  necessary  to  raise  the  water  to  any  proposed  height, 
must  at  least  be  capable  of  sustaining  in  equilibrium  the  pres- 
sure experienced  by  the  base  of  the  piston  when  it  is  kept  at 
rest,  and  the  fluid  has  attained  the  required  height.  This  pres- 
sure, then,  we  proceed  to  estimate. 

In  general  the  power  must  be,  at  least,  capable  of  sustaining 
the  weight  of  a  column  of  water  which  has  for  its  base  that  of 
the  piston,  and  for  its  altitude  the  distance  between  the  surface 
RS  of  the  water  in  the  reservoir  and  the  upper  surface  XY  of  that  Fig. 246. 
in  the  pump.  For  when  the  base  DC  of  the  piston  is  below  the 
surface  RS  of  the  water  in  the  reservoir,  it  is  manifest  that  the 
power  has  not  to  sustain  the  pressure  of  the  water  contained 
between  RS  and  DC;  because  this  pressure  is  counterbalanced 
by  that  of  the  water  surrounding  the  lower  part  of  the  pump, 
and  which  is  transmitted  by  means  of  the  inferior  orifice  of  the 
pipe.  The  power,  therefore,  has  only  to  sustain  the  pressure 
exerted  upon  the  surface  DC  by  the  fluid  comprehended  between 
RS  and  XF;  which  pressure  is  equal  to  the  weight  of  a  column 


406  Hydrodynamics. 

of  water  whose  base  is  CD  and  altitude  the  vertical  distance  be- 
414.     tween  RS  and  XY. 

When  the  piston  is  above  7?'  S',  the  surface  of  the  water  in 
the  reservoir,  then  it  is  evident  the  water  contained  between 
DC  and  R'  S'  docs  not  press  the  piston  downwards.  But,  as  in 
this  case  it  can  only  be  sustained  above  R'  S^  by  the  pressure  of 
the  air  upon  the  water  surrounding  the  pump,  and  as  this  pres- 
sure is  only  capable  of  sustaining  in  equilibrium  the  contrary 
pressure  of  the  air  upon  the  surface  XY,  it  follows  that  the  sur- 
face DC  \s  charged  with  a  weight  equivalent  to  the  column  which 
has  DC  for  its  base  and  CR'  for  its  altitude.  And  this  pressure, 
joined  to  that  which  is  exerted  upon  DC  by  the  superincumbent 
fluid  between  DC  and  XY,  makes  the  whole  pressure  upon  the 
piston,  as  before,  equal  to  thai  of  a  column  of  water  whose  base 
is  DC,  and  height  the  distance  between  XY  and  R'  S'. 

514.  The  sucking  pump  requires  in  its  theory  the  aid  of  other 
principles.  We  must  enquire  if  under  the  proposed  circumstan- 
ces the  water  can  possibly  be  raised  to  the  piston,  and  made  to 
pass  through  the  valve  L ;  for  in  some  cases  the  water  will  never 
pass  a  certain  altitude,  how  many  strokes  soever  we  give  to  the 
piston.  To  understand  this,  suppose  that  the  water  has  been 
Fig.  245.  actually  raised  to  T,  and  that  the  situation  of  the  piston  in  the 
figure  is  the  lowest  which  can  be  given  to  it;  and,  for  greater 
simplicity,  suppose  that  the  pump  is  of  the  same  internal  diame- 
ter throughout.  It  is  obvious  that  the  air  comprised  in  the  space 
CDTZ  is  of  the  same  density  and  elasticity  as  the  exterior  air  (at 
least  leaving  out  of  consideration  the  weight  of  the  valve  L,  and 
the  friction  attending  its  motion  ;)  for  if  its  spring  were  less,  the 
water  would  rise  higher  than  ZT,  and  if  it  were  greater,  it  would 
raise  the  valve  L,  and  mix  with  the  exterior  air  till  both  became 
of  the  same  density.  Suppose  now  that  the  play  of  the  piston, 
or  the  distance  through  which  it  is  raised  at  each  stroke,  is  DO ; 
then  when  the  base  CD  is  raised  to  OQ,  the  air  which  previously 
occupied  the  space  CDTZ  will  tend  to  expand  and  fill  the  space 
Q^OTZ ',  and,  if  the  water  did  not  rise,  would  actually  be  so  ex- 
panded. Its  elastic  force  would  then  be  less  than  that  of  the 
natural  air,  in  the  ratio  of  CDTZ  to  QOTZ,  or  of  DT  to  OT. 
468.    If,  therefore,  this  elastic  force,  together  with  the  weight  of  the  col- 


Water-Pumps.  407 

«mn  of  water  whose  height  is  ZR,  constitute  a  pressure  equal  to 
that  of  the  atmosphere,  or  equal  to  the  weight  of  a  column  of 
water  of  the  sauie  base,  and  at  a  mean  34  feet  in  height,  there 
will  be  an  equilibrium,  and  the  water  will  not  rise  further  ;  if  this  '^^^^ 
joint  pressure  is  greater  than  that  of  34  feet  of  water,  the  water 
cannot  be  retained  so  high  ;  but  if  it  is  less  than  a  column  of  34 
feet,  the  water  will  continue  to  rise  in  the  pump. 

515.  From  these  considerations  we  may  readily  investigate  a 
general  theorem. 

Let  a  be  the  altitude  or  vertical  distance  from  the  point  O  to 
the  surface  RS  of  the  water  in  the  reservoir,  p  =  OD,  the  play  of 
the  piston,  and  x  the  distance  OT ;  then  we  have  DT  =  x  — p, 
and  ST,  the  height  of  the  point  T,  will  be  «  —  x.  Since  the  air 
contained  in  CDTZ  has  the  same  density  and  elasticity  as  the  ''  ' 
exterior  air,  its  force  may  be  measured  by  a  column  of  water  of 
the  same  base  ZT  and  34  feet  high  ;  and  because  when  this  air 
is  so  expanded  as  to  fill  the  space  Q^OTZ,  the  elastic  force  will  be  i  ^ 
less  in  the  ratio  of  DT  to  OT,  we  shall  have  (rejecting  the  base  of  ^ 
the  column,  as  equally  affecting  every  part  of  the  process)  this  latter 
force  expressed  by  the  fourth  term  of  this  proportion, 

X  :  X  —  p  :  :  34   :  —  {x  —  p). 

But  the  force  which  the  water,  comprehended  between  ZT  and 
RS,  exerts  in  opposition  to  the  exterior  pressure  of  the  air,  is 
measured  by  the  height  a  —  x ;  consequently,  the  elastic  force 
of  the  air  in  the  space  Q^OTZ,  together  with  the  weight  of  the 
water  between  Z2^  and  RS,  will  be  expressed  by 

M{x—p)    , 

— 5^ i-^  4-  a  —  X. 

X 

Now  in  order  that  the  water  may  always  rise,  this  joint  pressure 
must  be  less  than  the  weight  of  a  column  of  water  34  feet  high 
by  some  variable  quantity,  which  we  will  call  y ;  so  that  die  fol- 
lowing equation  must  always  obtain,  namely, 

34  (z—;,)    , 

— ^— ; — ^-^  -f-  a  —  a?  =  34  —  y. 


/. 


408  Hydrodynamics. 

The  value  of  x   deduced   from  this  equation  is  ambiguous,  being 
thus  expressed ; 

a:  =  ^a+iy±s/  {{U-\-iyy  —  34/7). 

Now,  when  the  water  ceases  to  rise,  y  vanishes,  and  the 
equation  becomes  a:  =  ^  a  ±  y'  (i  a^  —  34  p) ;  of  which  the 
two  values  are  real,  so  long  as  ^  a^  is  greater  than  34  p.  Hence 
we  conclude,  that  luhen  one  fourth  of  the  square  of  the  greatest 
height  of  the  piston  above  the  surface  of  the  ivater  in  the  reservoir  is 
greater  than  34  times  the  play  of  the  piston,  there  are  always  two 
points  in  the  sucking  pump  where  the  water  may  stop  in  its  motion ; 
and  the  pump  must  be  reputed  bad  when  the  lowest  point  to 
which  the  piston  can  be  brought  is  found  between  these  two 
points. 

But  if  34  p  be  greater  than  ^  a^,  the  two  values  of  x,  when 
y  is  supposed  :=:  0,  become  imaginary  ;  so  that  in  a  pump  so 
constructed  it  is  impossible  that  y  should  vanish ;  that  is,  the 
pressure  of  the  exterior  air  always  prevails,  and  the  water  is 
not  arrested  in  its  passage.  Hence  we  conclude,  secondly,  that 
in  order  that  the  sucking  pump  may  infallibly  produce  its  effect,  the 
square  of  half  the  greatest  elevation  of  the  piston  above  the  water  in 
the  reservoir  must  always  be  less  than  34  times  the  play  of  the  piston. 

516.  This  general  rule  may  also  be  easily  deduced  geomet- 
245.  rically  ;  suppose  the  valve  E  to  be  placed  at  the  surface  RS  of 
the  water,  the  tube  to  be  of  a  uniform  bore,  and  YS  to  be  the 
height  of  a  column  of  water  whose  pressure  is  equal  to  that  of 
the  atmosphere;  that  is,  YS  =  34  feet.  Let  the  water  be 
raised  by  working  to  JV;  then  the  weight  of  the  column  of  water 
SJV,  together  with  the  elasticity  of  the  air  above  it,  exactly  balan- 
ces the  pressure  of  the  atmosphere  YS.  But  the  elasticity  of  the 
air  in  the  space  OM,  ( QO  being   the  highest  and    CD  the  lowest 

DN 

situation   of  the   piston.)   is  proportional  to   YS  .  Yf/ir''   ^"^'  ^*^"" 

sequently,  in  the  case  where  the  limit  obtains,  and  the  water  rises  no 
further,  we  shall  have  YS  =  NS  +  YS .  ^.   Transposing  NS, 

we  have 


Water-Pumps.  409 

DN 
ON' 


YS  —  NSov  YN=  YS.^; 


whence 

ON  :  DN  ::   YS  :   YN; 

or, 

ON—DNovDO  :  ON  :  :  YS  —  YNotNS  :  YS; 
consequently 

DO  .  YS  =  ON .  NS. 

Hence  we  see,  that  if  OS,  the  distance  of  the  piston  in  its  highest 
position  from  the  water,  and  DO  the  length  of  the  half-stroke,  or 
the  play  of  the  piston,  be  given,  there  is  a  certain  determinate 
height,  as  SN,  to  which  the  water  can  be  raised  by  the  difference 
of  the  pressures  of  the  exterior  and  interior  air ;  for  YS  is  to  be 
considered  as  a  constant  quantity,  and,  of  course,  when  DO  is 
given,  ON .  NS  is  given  likewise.  To  ensure,  therefore,  the  de- 
livery of  water  by  the  pump,  the  stroke  must  be  such  that  the 
rectangle  DO  .  YS  shall  be  greater  than  any  rectangle  that  can 
be  made  of  the  parts  of  OS ;  that  is,  greater  than  the  square  of 
\  OS,  by  a  well-known  theorem. 

Hence  we  deduce  a  practical  maxim  of  the  same  import  as 
the  preceding,  which  is,  that  no  sucking  pump  can  raise  water  ef- 
fectually, unless  the  play  of  the  piston  in  feet  be  greater  than  the 
square  of  the  greatest  height  of  the  piston,  divided  by  136. 

517.  Resuming  the  equation 

U{x—p)    ,  ,,. 

and  finding  thence  the  value  of  y,  we  have 
T-  —  ff  I  -f  34  » 

^  X 

Now  let  AB  represent  the   greatest  height  of  the  piston  above  the  Fig.  248, 
surface  of  the  water  in  the  reservoir,  and  AD  the   play  of  the  pis- 
Mech.  52 


410  Hydrodynamics. 

ton  ;  suppose  the  different  portions  AP  of  the  line  AB  to  represent 
the  successive  vahies  of  x,  and  lay  down  upon  the  perpendicu- 
lars PMthe  values  of  y  which  correspond  to  these  assumed  values 
of  jr ;  so  shall  we  have  a  curve  MMC,  which,  while  i  a^  is  greater 
Fig.248.  than  24 p,  will  cut  AB  in  two  points  /and  /',  in  such  a  manner 
that  the  ordinates  PM  will  lie  on  different  sides  of  AB  ;  the  ordi- 
nates  which  are  on  the  right  AB  showing  the  positive  values  of  y, 
and  those  which  are  on  the  left  AB  the  negative  values.  We  see, 
therefore,  that  so  long  as  -  or  is  greater  than  34  p,  the  pressure  of 
the  exterior  air  is  strongest,  until  the  water  has  attained  the  height 
BI'.  At  this  point  T,  it  will  stop  (the  motion  acquired  being  left  out 
of  consideration,)  because  the  value  of  y  is  :=  0.  But  if  the  water 
by  the  motion  it  has  acquired  continues  to  rise  till  it  reaches  some 
point  between  /'  and  /,  it  will  not  stop  there,  but  will  descend,  if  the 
valve  does  not  oppose  its  descending  motion  ;  because  the  value  oi  y, 
being  there  negative,  indicates  that  the  pressure  of  the  exterior  air  is 
weaker  than  the  united  pressures  of  the  water  and  the  interior  air. 
If  the  water  reaches  the  point  /,  it  will  stop  there,  for  the  same 
reason  that  it  would  at  the  point  I' ;  but  if  it  rises  above  J, 
there  is  then  no  reason  to  fear  that  it  will  descend  ;  for  all  the  ordi- 
nates PM  between  /and  A  being  positive,  show  that  in  that  portion 
of  the  pump  the  pressure  of  the  exterior  air  exceeds  the  combined 
efforts  of  the  interior  air  and  water. 

518.  When,  on  the  contrary,  the  value  of  \  a^  is  less  than  that 

Fig.249.  of  34  p,  the  curve  will  not  intersect  the  axis  AB  ;  all  the  ordinates 

are  positive,   and  consequently  the  pressure  of  the  exterior  air  is 

always  the  strongest.     This  confirms  and  illustrates  what  has  been 

laid  down  in  article  515. 

If  the  sucking  pump  were  to  be  placed  so  high  above  the 
usual  surface  of  the  earth  (as  at  the  top  of  a  high  mountain),  or 
so  low  beneath  it  (as  in  a  deep  mine),  that  the  pressure  of  the 
atmosphere  would  be  sensibly  different  from  the  assumed  mean 
pressure  equivalent  to  34  feet  of  water,  we  must  then  in  all  the 
preceding  investigation  change  the  co-efficient  34  to  that  which 
would  express  the  height  in  feet  of  the  corresponding  column  of 
water.  And  these  equivalent  columns  may  always  be  ascertain- 
ed by  means  of  the  height  of  the  mercurial  column  in  the  barora- 


Water-Pumps.  411 

eter  ;  the  proportion  being  this  ;  as  30  inches,  the  mean  altitude  of 
the  mercurial  column,  is  to  34  feet,  the  mean  height  of  the  column 
of  water,  so  is  any  other  mercurial  column  in  inches  to  its  corre- 
sponding column  of  water  in  feet. 

519.  In  the  preceding  calculation  the  pump  has  been  supposed 
of  a  uniform  bore  throughout ;  when  this  is  not  the  case  the  so- 
lution is  rendered  somewhat  more  complex,  but  not  difficult.  To 
calculate  the  effort  of  the  interior  air  when  the  water  has  not 
reached  the  body  of  the  pump,  having  only  attained  the  height 
jHJV*,  for  example,  we  must  use  this  proportion  ;  as  the  space  Fig.24.5. 
qOVJVMl  :  the  space  CDVJVMl  :  :  34  feet  :  a  fourth  term, 
which  being  added  to  the  weight  of  the  column  of  water  whose 
height  is  JYH,  ought  again  to  be  equal  to  34  —  y,  as  before.  Be- 
sides, when  the  sucking   pipe  FG  is  of  a  smaller  diameter  than  the 

body  of  the  pump,  if  the  conditions  which  we  have  before  specified 
obtain,  the  pump  cannot  fail  to  produce  the  proper  effect  ;  for 
the  air  is  dilated  with  more  facility  in  this  latter  case  than  when 
the  whole  is  of  the  same  diameter.  We  need  only  add  on  this 
point,  that  if  the  length  of  the  stroke  in  a  uniform  pump,  which 
is  requisite  to  render  the  machine  effectual,  be  greater  than  can 
conveniently  be  made,  it  may  be  diminished  by  contracting  the 
diameter  of  the  sucking  pipe  in  the  subdupUcate  ratio  of  the  diminu- 
tion of  the  length  of  the  stroke. 

520.  As  to  the  effort  of  which  the  power  ought  to  be  capable 
to  sustain  the  water  at  a  determinate  height  YH,  it  will  be  measured 
according  to  what  we  have  said  respecting  the  lifting  pump  by  the 
weight  of  a  column  of  water  whose  base  is  equal  to  CD,  and  height 
that  of  XY  above  RS.  Here,  too,  we  leave  out  of  considera- 
tion the  friction  and  the  weight  of  the  piston. 

521.  The  velocity  of  the  water  flowing  from  the  sucking 
pipe  into  the  barrel  should  be  equal  to  the  velocity  with  which 
the  piston  moves.  For  if  it  be  greater,  less  work  will  be  done 
than  the  pump  is  competent  to  effect ;  and  if  it  be  less,  a  vacuum 
will  be  produced  below  the  piston,  which  will  therefore  be  moved 
upwards  with  great  difficulty. 


412  Hydrodynamics, 


Of  the  Syphon. 

Fig. 250.  522.  If  we  introduce  into  a  vessel  of  water  or  other  liquid, 
the  shorter  branch  of  a  recurved  tube  DEF,  called  a  syphon,  and 
exhaust  the  air  from  this  tube  by  the  mouth  or  otherwise,  the 
water  will  rise  in  the  tube  and  flow  out  at  F,  until  the  surface  of 
the  fluid  in  the  vessel  shall  have  descended  to  the  opening  D  of  the 
syphon. 

The  reason  of  this  phenomenon  is,  that  when  the  contained 
air  is  withdrawn,  the  pressure  of  the  interior  air  upon  the  surface 
AB  causes  the  fluid  to  rise  into  the  syphon  and  to  flow  through 
the  branch  EF.  And  although  when  the  flowing  has  commenced, 
the  air  presses  the  fluid  at  the  point  F  with  a  force  very  nearly 
equal  to  that  which  is  exerted  upon  the  surface  of  the  water  ia 
the  vessel,  still  a  transverse  lamina  at  F  is  urged  downward  by  the 
entire  column  of  water  ZF;  this  column  must  therefore  descend; 
but  in  descending  it  tends  to  form  a  vacuum  at  /,  which  cannot  fail 
of  being  filled  by  the  action,  always  exerted,  of  the  incumbent  air 
upon  the  surface  of  the  fluid  in  the  vessel. 

It  will  hence  be  seen  that  during  the  discharge  through  the 
syphon  the  air  acts  only  with  an  effort  proportional  to  the  differ- 
ence of  level  IF  between  F  and  the  surface  of  the  water  in  the 
vessel ;  so  that  the  flowing  will  be  so  much  the  more  rapid 
according  to  the  difference  of  the  two  branches  of  the  syphon  ; 
thus  if  F  and  D  were  on  a  level,  no  motion  of  the  fluid  would  take 
place.  We  say  generally,  that  the  branch  EF  must  be  longer 
than  the  branch  ED ;  but  in  using  this  language  it  must  be  under- 
stood that  the  vertical  height  of  E  above  F  must  be  greater  than 
that  of  E  above  D.  The  absolute  height  is  not  concerned,  DE  may 
be  rendered  much  longer  than  DF  by  being  made  crooked  ;  so  long 
as  the  point  D  is  higher  than  F,  the  fluid  will  pass  until  it  arrives 
at  D  provided  the  height  of  E  above  D  does  not  exceed  34  feet. 


Steam-Engine.  413 


Of  the  Steam-Ejigine. 

523.  The  whole  theory  of  the  steam  engine  is  founded  upon 
two  principles,  the  developement  of  the  elastic  force  of  aqueous 
vapor  by  heat,  and  the  sudden  precipitation  of  this  vapor  by 
cooling.  On  account  of  the  extensive  uses  of  this  machine  in  the 
arts,  we  shall  here  treat  of  it  at  some  length. 

Although  it  is  generally  sufficient  in  mechanics  to  create  any 
one  force  or  motion,  in  order  to  be  able  thence  to  deduce  all 
sorts  of  motions,  yet  for  the  sake  of  distinctness  we  shall  suppose 
that  it  is  proposed,  in  the  first  place,  to  draw  water  from  a  mine 
by  means  of  a  sucking  pump  T'  T".  Here  the  point  in  question  isFig.25l. 
to  raise  the  piston  P'.  For  this  purpose  we  attach  the  piston  rod 
to  a  chain  applied  to  one  of  the  extremities  A'  of  a  bent  lever 
moving  about  its  centre  C.  It  is  evident,  that  if  we  attach  to  the 
opposite  arm  of  the  lever  a  similar  chain  represented  by  AD,  we 
shall  only  have  to  pull  this  chain,  in  order  to  raise  the  piston  P', 
and  draw  the  water  into  the  body  of  the  pump  by  the  external 
pressure  of  the  atmosphere.  This  being  done,  the  valves  placed 
at  the  bottom  of  the  pump  will  close  ;  and  the  apparatus  being 
left  to  itself,  if  we  suppose  the  weight  of  the  piston  P',  together 
with  that  of  the  frame  which  supports  it,  to  exceed  the  total 
weight  of  AD  and  P,  it  is  evident  that  the  piston  P'  will  descend 
into  the  water  by  its  own  weight  and  cause  the  water  to  raise 
the  valve  opening  through  its  centre ;  and  having  reached  the 
bottom  of  the  body  of  the  pump,  will  separate  this  water  entirely 
from  the  water  below.  Then  by  pulling  anew  the  chain  AD,  we 
shall  raise  this  water  with  the  piston,  and  at  the  same  time  draw 
more  water  into  the  body  of  the  pump  ;  after  which  the  piston 
will  descend  by  its  own  weight  in  the  same  manner  as  before, 
and  so  on  indefinitely.  It  remains  then  to  give  the  requisite  mo- 
tion to  the  chain  AD.  For  this  purpose  we  attach  its  lower  ex- 
tremity D  to  another  piston  P,  moving  like  the  first  in  the  body  of 
the  pump   TT  likewise   cylindrical ;   but  suppose  the  bottom  of 


414  Hydrodynamics. 

the  body  of  the  pump,  instead  of  being  immersed  in  water,  to 
communicate  with  an  air  pump  by  means  of  which  the  air  in  it 
can  be  exhausted.  It  is  manifest  that  after  the  exhaustion  the 
pressure  of  the  atmosphere  upon  the  upper  surface  of  the  piston 
P  will  tend  to  make  it  descend  ;  and  will,  in  fact,  make  it  descend, 
if  the  whole  effect  of  this  pressure  exceed  the  weight  of  the  pis- 
ton P',  together  with  that  of  the  column  of  water  to  be  raised. 
Now  the  piston  P,  having  descended  to  the  bottom  of  the  body 
of  the  pump,  let  the  air  be  admitted  below ;  then  the  pressure  on 
the  two  surfaces  will  be  equal  ;  and  the  excess  of  the  weight  of 
the  piston  P'  beginning  to  act,  P  will  rise  in  the  bore  of  the  pump  ; 
after  which,  if  we  make  a  new  exhaustion  under  P,  we  shall 
cause  P  to  descend  and  P'  to  rise,  and  we  can  repeat  these  mo- 
tions at  pleasure. 

But  it  will  be  seen  that  the  air  pump  could  hardly  be  em- 
ployed upon  so  large  a  scale.  To  supply  its  place  we  introduce 
steam  into  the  body  of  the  pump  TT.  The  simplest  method  of 
doing  this,  and  which,  (although  it  has  not  hitherto  been  most 
generally  employed,)  is  nevertheless  attended  with  some  pecu- 
Fig.  252.  liar  advantages,  is  the  following.  Under  the  body  of  the  pump 
TT,  let  there  be  placed  a  boiler  FF,  filled  in  part  with  boiling 
water,  the  steam  of  which  being  equal  or  a  very  little  superior  in 
elasticity  to  the  pressure  of  the  atmosphere,  may  be  introduced 
at  pleasure  into  the  cylinder  TT  by  turning  a  stop-cock  R,  which 
opens  a  communication  between  the  pump  and  the  boiler.  Let 
there  be  likewise  at  the  bottom  of  the  cylinder  a  small  lateral 
passage  VS,  shut  by  a  valve  S  opening  outwards.  Now  the  pis- 
ton P  being  forced  to  the  bottom  or  nearly  to  the  bottom  of  the 
cylinder  TT,  and  the  space  below  being  filled  with  air,  turn  the 
stop-cock  R  which  communicates  with  the  boiler.  The  steam 
will  rush  into  the  cylinder,  and  by  its  impulse,  together  with  its 
elastic  force,  will  in  part  expel  the  air  remaining  in  the  cylinder 
by  forcing  it  to  open  the  valve  S.  In  this  operation  a  great  quan- 
tity of  steam  is  suddenly  condensed  by  the  cold  surface  of  the 
cylinder  TT  and  that  of  the  piston  P,  and  being  reduced  to  water, 
is  made  to  pass  out  through  a  descending  tube  EGS',  recurved  at 
the  lower  end  and  terminated  by  a  valve  S',  opening  outwards. 
This   condensation,    and   consequent   loss  of   steam   produced   by 


Steam  Engine.  415 

cooling,  will  continue  until  the  piston  and  the  portion  of  the  cyl- 
inder situated  between  it  and  the  boiler,  are  brought  to  the  tem- 
perature of  the  steam  itself.  Then,  the  steam  preserving  its  elas- 
tic form  under  the  piston  P,  counterbalances  entirely  or  partly 
the  pressure  of  the  atmosphere  upon  its  upper  surface.  The 
excess  of  weight  in  P',  acting  therefore  without  obstacle,  causes 
A'P'  to  descend  and  AP  to  ascend,  which  tends  to  produce  in  the 
cylinder  a  vacuum  into  which  steam,  rising  from  the  boiler,  con- 
tinues to  enter,  until  the  piston  P,  having  reached  its  highest  point, 
the  cylinder  is  completely  filled  with  steam.  Having  obtained 
this  limit,  the  steam  opens  the  valve  <S  and  escapes,  at  first  slowly 
and  in  the  form  of  a  cloud,  on  account  of  its  mixing  with  the  air 
and  drops  of  water.  According  as  the  air  is  expelled  this  cur- 
rent becomes  gradually  stronger  and  more  transparent.  When 
the  operator  perceives  that  this  point  is  attained,  he  turns  back 
the  stop-cock,  and  then  the  whole  cavity  of  the  pump  remains 
filled  with  pure  steam  which  only  wants  to  be  condensed  by 
sudden  cooling  in  order  to  leave  a  vacuum  under  the  piston  P. 
This  condensation  is  effected  by  the  introduction  of  cold  water 
which  is  made  to  descend  from  an  elevated  reservoir  Z  through 
the  tube  ZR'I,  closed  at  R'  by  a  stop-cock  called  the  injection 
stop-cock.  Upon  turning  this,  the  cold  water  thrown  into  the 
cylinder  TT,  precipitates  entirely  or  partly  the  steam  contained 
there  and  it  flows  out  through  the  tube  EGS'  with  the  water  which 
results  from  this  condensation  ;  then  a  vacuum  being  left  under 
the  piston  P,  the  pressure  of  the  atmosphere  causes  it  to  descend. 
This  piston  is  again  raised  by  the  introduction  of  fresh  steam ; 
for  if,  as  we  have  supposed,  the  water  in  the  boiler  is  kept  in  a 
state  of  ebullition,  the  steam  has  an  elastic  force  at  least  equal  to 
that  of  the  air,  and  consequently  its  introduction  under  the  piston 
P  is  sufficient  to  counteract  the  pressure  of  the  atmosphere  ;  so 
that  the  excess  of  weight  in  P'  will  raise  P  as  before.  But  on 
the  other  hand,  the  steam,  if  heated  too  much,  may  by  its  elastic 
force  cause  the  boiler  to  burst.  To  guard  against  this,  we  adapt 
to  the  top  of  the  boiler  a  safety  valve  S'',  which  opens  outwards 
with  a  known  effort.  When  the  elastic  force  of  the  steam  is 
equal  or  inferior  to  that  of  the  external  air,  the  valve  remains 
closed  ;  but  when  it  becomes  equal  to  that  of  the  atmosphere 
and  the  rcsislaucc  of  the  valve  together,  the  steam  escapes  and 


416  Hydrodynamics. 

no  explosion  is  to  be  feared.  Still,  however,  it  is  necessary  that 
the  boiler  should  be  made  stronger  than  this  limit  of  resistance 
supposes.  For  when  the  steam  rushes  into  the  cold  cylinder 
and  is  condensed  there,  this  precipitation  is  so  rapid  that  the 
new  steam  formed  at  the  same  time  in  the  boiler  is  not  always 
sufficiently  instantaneous  to  supply  its  place.  For  a  moment  a 
vacuum  is  left  in  the  boiler,  and  the  pressure  of  the  external 
air  being  no  longer  counterbalanced,  the  boiler  may  burst  inward 
if  its  sides  are  not  sufficiently  thick.  This  accident  sometimes 
happens,  but  it  may  always  be  prevented  by  means  of  a  second 
safety  valve,  which  opens  inward  whenever  the  external  pressure 
becomes  too  great. 

From  this  explanation  it  would  seem  that  when  the  engine 
was  once  in  operation  nothing  would  remain  in  the  piston  or  body 
of  the  pump  but  pure  steam  or  a  vacuum.  But  it  must  be  re- 
marked that  the  injected  water  has  also  some  air  combined  with 
it  which  escapes  into  the  body  of  the  pump  ;  since  it  is  found  there 
in  a  highly  rarified  state,  being  heated  to  a  considerable  degree  by 
the  great  quantity  of  heat  disengaged  during  the  condensation. 
Happily  this  air,  being  in  small  quantity,  and  contained  in  a  small 
space,  is  easily  expelled  through  the  valve  S  by  the  first  effort  of 
the  steam  introduced  into  the  cylinder. 

524.  The  apparatus  which  we  have  described  is  not  precisely 
the  one  first  invented.  It  appears  that  the  original  attempt  was 
simply  to  employ  the  force  of  steam  as  a  moving  power.  But  the 
more  ingenious  discovery  of  the  method  of  condensing  the  steam 
by  cooling  was  not  made  until  1696  ;  and  the  English  attribute 
it  to  Capt.  Savary,  who  published  an  account  of  it  in  a  treatise 
entitled  The  Miner^s  Friend.  The  mode  of  applying  this  princi- 
ple was  still  very  imperfect.  In  1705,  Newcomen,  another  Eng- 
lishman, gave  it  the  form  which  we  have  described,  in  which,  under 
the  name  of  the  atmospherical  engine,  it  was  a  long  time  not  unprofi- 
tably  employed. 

Nevertheless,  with  the  progress  we  have  made  in  mechanics 
and  the  natural  sciences,  it  is  easy  to  perceive  that  this  engine 
had  many  theoretical  defects.  It  was  a  great  imperfection  that  it 
required  an  intelligent  person  to  watch  it  for  the  purpose  of  turn- 
ing the  stop-cocks  to  introduce  water  and  steam  every  time  the 


Sieam-Engine.  417 

piston  reached  its  limit.  A  good  machine  ought  always  to  tend 
itself  by  means  of  the  first  moving  power,  without  any  foreign 
aid.  Another  great  inconvenience  was  the  introduction  of  steam 
into  a  cold  cylinder ;  since  a  great  loss  was  thereby  occasioned, 
and  was  repeated  at  each  stroke  of  the  piston,  the  cylinder  being 
continually  cooled  by  the  injected  water  necessary  for  the  precip- 
itation. But  these  defects,  which  in  the  present  state  of  the  sci- 
ences we  are  so  prompt  to  observe,  could  not  at  first  have  been  so 
easily  detected.  They  were  perceived  and  corrected  in  1764,  by 
Watt,  the  disciple  and  friend  of  Black.  Being  then  at  Glasgow, 
where  he  was  employed  as  a  mathematical  instrument  maker,  he 
was  directed  to  repair  a  small  model  of  Newcomen's  engine, 
which  belonged  to  the  University  in  that  city.  In  the  course  of 
his  attempts  to  make  its  operation  satisfactory,  he  observed  that 
it  consumed  more  coal  in  proportion  to  its  size  than  the  large 
engines.  Being  curious  to  ascertain  the  cause  of  this  dif- 
ference, and  wishing  to  remedy  so  great  a  defect,  Watt  made 
numerous  experiments  for  the  purpose  of  determining  what  sub- 
stance is  the  most  suitable  for  the  cylinders  ;  and  what  are  the 
most  proper  means  of  creating  a  perfect  vacuum ;  what  is  the 
temperature  to  which  water  rises  in  boiling,  under  different 
pressures;  and  what  the  quantity  of  water  necessary  to  produce 
a  given  volume  of  steam,  under  the  ordinary  pressure  of  the  at- 
mosphere. He  determined  also  the  precise  quantity  of  coal 
necessary  to  convert  a  known  weight  of  water  into  vapor,  and 
the  quantity  of  cold  water  required  to  precipitate  a  given  weight 
of  steam.  These  several  points  being  once  exactly  ascertained, 
he  was  led  to  perceive  the  defects  of  Newcomen's  engine,  and 
to  assign  the  cause  of  these  defects.  He  saw  that  the  steam 
could  not  be  condensed  so  as  to  produce  any  thing  like  a  va- 
cuum, unless  the  cylinder  and  the  water  it  contained,  as  well  as 
that  injected  and  that  arising  from  the  condensed  steam,  were 
cooled  down  at  least  to  the  temperature  of  about  33°,  and  that 
at  a  higher  temperature  the  steam  had  still  an  elasticity  strong 
enough  to  oppose  a  very  perceptible  resistance  to  die  weight  of 
the  atmosphere.  On  the  other  hand,  when  it  is  proposed  to  at- 
tain more  perfect  degrees  of  exhaustion,  the  requisite  quantity  of 
injected  water  is  augmented  in  a  very  rapid  proportion;  and 
hence  results  a  great  loss  of  steam,  when  the  cylinder  is  again 
filled.  These  facts  led  Watt  to  conclude,  that,  in  order  to  eflect 
Mech.  53 


418  Hydrodynamics . 

tlie  most  coin[)lete  vacuum  possible,  with  the  least  expense  of 
steam,  it  was  necessary  that  the  cylinder  should  be  kept  con- 
stantly as  hot  as  the  steam  itself,  and  that  the  injection  of  cold 
water  should  take  place  in  a  separate  vessel  which  he  called  the 
condenser,  and  whose  communication  with  the  cylinder  was  sud- 
denly opened  at  the  moment  of  the  injection.  Indeed  after  what 
is  now  known  respecting  the  equilibrium  of  fluids,  it  is  manifest 
that,  if  the  air  be  exhausted  from  the  condenser,  the  steam  from 
the  cylinder  will  enter  it,  on  account  of  its  own  elasticity,  the 
instant  a  communication  is  opened  ;  and  an  injection  of  wa- 
ter made  at  this  instant,  will  precipitate  not  only  the  steam 
actually  in  the  condenser  but  also  on  the  same  principle, 
all  the  steam  contained  in  the  cylinder,  which,  rushing  into 
the  vacuum,  continually  formed  by  precipitation  in  the  conden- 
ser, is  converted  almost  instantaneously  into  water.  It  only 
remains,  then,  to  remove  this  water  and  disengage  the  air,  in 
order  to  preserve  always  a  vacuum  in  the  condenser.  Watt 
constructed  a  pump  in  such  a  manner  as  to  be  moved  by  the 
engine  itself  and  which  played  continually  in  a  lube  void  of  air, 
the  lower  part  being  immersed  in  the  water  of  the  condenser. 
Finally,  the  condition  of  keeping  the  cylinder  hot  could  not  be 
fulfilled  while  there  was  a  free  admission  of  atmospheric  air  to 
the  interior  of  its  upper  surface,  which  in  the  apparatus  of  New- 
comen,  caused  the  piston  to  descend  ;  especially,  since  in  order 
to  prevent  the  passage  of  the  steam  between  the  cylinder  and 
piston,  the  latter  was  ordinarily  covered  with  a  stratum  of  cold 
water  which  kept  the  interior  of  the  cylinder  wet.  Watt  con- 
ceived the  bold  and  ingenious  idea  of  dispensing  entirely  with  the 
pressure  of  the  atmosphere,  and  making  the  piston  descend  by 
the  force  of  steam  alone,  by  introducing  it  alternately  above  and 
below,  and  causing  at  the  same  time  a  vacuum  in  each  case  in 
the  manner  already  described.  Then  he  enclosed  the  rod  of 
his  piston  in  a  collar  of  leather  to  prevent  all  access  of  air  to 
the  interior  of  the  cylinder;  and  employing  steam  of  an  elas- 
ticity equal  or  even  a  litdc  superior  to  the  pressure  of  the  at- 
mosphere, he  obtained  alternately  above  and  below  the  piston  a 
force  equal,  or  a  litde  superior,  to  the  atmospheric  pressure.  He 
was  then  able,  by  substituting  stiff  rods  for  the  chains  AP,  A'P', 
to  produce  a  force  in  both  directions  ;  whereas,  in  Newcomen's 
engine,  the  time  during  which  the   piston  was  ascending  in  the  cyl- 


Steam-Engine.  419 

inder,  was  entirely  lost  so  far  as  the  steam  was  concerned,  since 
it  was  raised  simply  by  the  excess  of  weight  on  the  other  arm  of 
the  large  lever.  Here  was  a  saving  both  of  time  and  expense  j 
for  the  piston  was  accelerated  each  way  by  steam,  and  the 
quantity  of  fuel  employed  to  keep  it  hot,  during  its  ascent,  was 
not  wasted.  Watt  took  care,  moreover,  to  surround  the  cylin- 
der with  a  case  of  wood,  or  some  other  substance  which  is  a  non- 
conductor of  heat ;  into  the  interior  of  which,  he  also  occa- 
sionally introduced  the  steam  as  a  means  of  keeping  it  warm. 
He  likewise  used  so  much  economy  in  the  construction  of  the 
different  parts  of  his  engine,  that  he  succeeded  in  saving  two 
thirds  of  the  steam  employed  by  Newcomen.  The  steam- 
engine,  thus  improved,  is  represented  in  figure  253,  the  ex- 
planation of  which  will  now  be  easily  understood.  FD  is 
the  boiler,  in  which  the  water  is  converted  into  steam  by  the 
heat  of  the  furnace  below.  This  boiler  is  sometimes  made 
of  copper,  but  more  frequently  of  iron.  Its  bottom  is  con- 
cave and  the  flame  curls  around  it.  Towards  the  top,  it 
has  a  safety  valve  S"  fitted  to  resist  a  greater  or  less  effort  before 
opening,  according  to  the  degree  of  elastic  force  to  be  em- 
ployed. That  the  conversion  of  water  into  steam  may  be  con- 
stant, it  is  necessary  that  the  water  in  the  boiler  should  be  kept 
always  at  the  same  level,  and  consequently  that  it  should  be 
supplied  as  fast  as  it  is  evaporized.  This  is  effected  by  a  tube 
V  V,  which  supplies  the  boiler  from  a  small  reservoir  z,  filled 
with  water,  already  heated,  which  the  pump  t  t  takes  from  the 
condenser  and  forces  into  the  lateral  pipe  t'  t'.  But  in  order 
to  introduce  this  into  the  boiler  only  when  it  becomes  necessary, 
the  upper  orifice  of  the  tube  v  v'xs  closed  by  a  stopper,  which  is 
raised  or  lowered  by  means  of  a  small  lever  a  h  ;  and  at  the 
other  arm  b  of  the  lever  hangs  a  wire  h  m,  drawn  downwards  by 
a  weight  m,  which  is  so  adjusted  in  the  boiler  as  to  keep  on  a 
level  with  the  upper  surface  of  the  water.  Then  if  the  water 
falls  below  this  level,  the  weight  m,  which  it  supports  in  part,  will 
descend  with  it  ;  the  lever  turning  will  raise  the  stopper,  and 
suffer  the  water  to  pass  into  the  boiler  ;  but  as  soon  as  the  level 
is  reestablished,  the  lever  a  h  will  again  become  horizontal,  and 
return  the  stopper  to  its  place.  From  the  top  of  the  boiler  pro- 
ceeds the  steam  tube  W^  which  conveys  the  steam  to  the  top  of 
the  cylinder    TT,  by  the    valve   (S,    and  to  the   bottom,    by   the 


420  Hydrodynamics. 

valve  S\  The  communication  between  S  and  S'  is  effected  by 
means  of  a  curved  pipe,  whose  plane  of  curvature  is  perpendic- 
ular to  the  plane  of  projection  in  figure  253.  One  j)art  only  of 
this  pipe  is  represented  in  this  figure  ;  in  order  that  the  two  valves 
S",  S'",  may  be  exhibited,  of  which  we  are  presently  to  speak ; 
but  the  whole  may  be  seen  in  figure  254,  where  it  is  presented  in 
profile.  The  valves  S'',  S'",  are  those  by  vvhich  the  steam  of  the 
cylinder  is  brought  into  communicaUon  with  the  condenser,  on 
both  sides  of  the  piston  ;  and  they  are  opened  and  closed  at  proper 
times  by  the  engine  itself,  with  the  aid  of  two  small  projections, 
1,  2,  attached  to  the  rod  t  t  of  the  pump  which  serves  to  exhaust 
the  condenser  C  This  opening  is  effected  a  little  before  the 
piston  has  completed  its  verfical  motion,  and  the  communicaUon 
is  then  established  between  the  two  surfaces,  in  order  that  the 
equality  of  pressure  thence  resulting  may  weaken  the  effort  vvhich 
would  be  made  if  it  were  exerted  on  one  side  only,  and  prevent  the 
sudden  jar  which  would  follow  from  the  piston's  striking  with  its  full 
force  against  the  bottom  of  the  cylinder. 

These  are  the  principal  conditions  relating  to  the  action  of 
the  steam,  but  there  are  others  which  relate  to  the  manner  of 
directing  this  action.  Indeed,  a  bare  inspection  of  the  figure 
will  show  that  the  rod  of  the  great  piston  and  that  of  the  pump 
which  exhausts  the  condenser,  being  both  inflexible,  cannot  be 
attached  immediately  to  the  large  lever  AB ;  for  each  point  of 
this  lever,  describing  an  arc  of  a  circle  about  its  centre  of  rota- 
tion, would  tend  to  change  the  point  of  attachment  from  a  ver- 
tical direction,  and  this  effort  would  break  the  engine.  It  is  on 
this  account,  that,  in  the  engine  of  Newcomen,  where  the  piston 
acted  only  in  its  descent,  its  communication  with  the  large  lever 
was  made  by  a  chain  applied  to  the  arc  of  a  circle.  But  in  the 
engine  under  consideration,  the  inflexibility  of  the  rods  requires 
some  other  mode  of  communication.  Mr.  Watt  effected  this  by 
means  of  a  particular  assemblage  of  metallic  bars,  moving  upon 
one  another,  and  which  compensate  by  their  action,  for  the  want 
of  perfect  verficality  in  the  motion  of  the  large  lever.  The  figure 
represents  also,  several  other  useful  appendages  to  the  engine, 
such  as  flies  to  regulate  its  motion,  and  wheels  to  transmit  it. 
There  is  also  a  very  essential  part  designated  by  G,  and  'Galled 
the  governor.    It  consists  of  a  vertical  rod  which  is  kept  contin- 


Steam-Engine.  421 

ually  in  rotation  by  the  engine  itself,  and  which  carries  at  its 
summit  a  parallelogram  formed  of  metallic  plates,  turning  free- 
ly one  upon  the  other,  in  such  a  manner  that  this  parallelogram 
may  open  more  or  less  in  a  horizontal  direction,  according  as 
the  plates  diverge  more  or  less  from  the  axis.  This  diverg- 
ence is  produced  by  the  centrifugal  force  exerted  by  the  axis 
in  turning,  under  the  influence  of  the  engine,  with  greater  or  less 
rapidity  ;  which  causes  the  upper  vertex  of  the  parallelogram 
to  be  depressed  when  the  engine  moves  more  rapidly,  and  ele- 
vated when  it  moves  more  moderately.  In  order  to  give  great- 
er force  to  this  ascending  and  descending  motion,  the  extremi- 
ties of  the  plates  are  loaded  with  spheres  of  solid  metal,  and 
exert  their  power  upon  a  lever,  whose  other  branch  communi- 
cates with  a  plate  placed  transversely  in  the  passage  through 
which  the  steam  passes  from  the  boiler  to  the  cylinder ;  so  that, 
when  the  engine  works  too  slowly,  the  plate  turns  in  such  a 
manner  as  to  give  a  freer  passage  to  the  steam ;  and,  on  the 
other  hand,  if  the  engine  moves  too  rapidly,  the  plate  takes  a 
position  more  nearly  in  a  transverse  direction,  and  diminishes 
the  passage  through  which  the  steam  has  to  pass.  Thus  the 
engine  is  made  to  govern  and  regulate  itself,  in  such  a  manner 
as  to  preserve  in  its  motions  that  uniformity  which  its  purposes 
require.  There  are  many  other  details  which  would  furnish 
matter  for  curious  speculation  ;  but  these  details,  belonging  to 
the  mechanism,  must  be  omitted  here,  to  make  room  for  some 
other  particulars,  no  less  important,  relating  to  the  principles  of 
the  machine. 

525.  The  most  essential  of  these  is  the  determination  of  the 
temperature,  at  which  it  is  most  advantageous  to  employ  the 
steam.  In  fact,  the  higher  the  temperature  is,  the  greater  will 
be  its  elastic  force,  and  consequently  the  greater  will  be  its 
effect  upon  the  surface  of  the  piston,  the  vacuum  being  always 
on  the  other  side.  From  the  experiments  of  Southern,  Clement, 
Desormes,  and  Despretz,  it  has  been  found  that  the  total  quantity 
of  heat  necessary  to  change  the  same  mass  from  water  to  a  state 
of  vapor,  is  very  nearly  if  not  quite  the  same  for  all  tempera- 
tures. According  to  this  principle,  then,  it  will  not  be  necessary 
to  consume  more  fuel  in  order  to  form  a  given  weight  of  steam 
of    a    higher    temperature    and    more   elastic,   than    is    required 


422  Hydrodynamics. 

for  the  same  weight  at  a  lower  temperature  and  less  elastic. 
But  while  the  steam  is  raised  to  a  higher  temperature,  it  becomes 
also  more  dense  ;  so  that  considerably  more  weight  is  necessary 
to  fill  the  same  cylinder,  than  when  its  temperature  is  lower. 
This  second  circumstance,  therefore,  requires  in  the  same  ma- 
chine, a  greater  quantity  of  fuel,  according  as  the  temperature 
of  the  steam  is  increased  ;  so  that  it  only  remains  to  ascertain 
whether  this  increase  of  fuel  is,  or  is  not,  compensated  by  a 
corresponding  increase  of  elastic  force.  Now  this  point  is  easi- 
ly decided,  if  we  reflect  that  the  vapor  of  water  and  that  of 
odier  liquids,  so  long  as  they  exist  in  a  state  of  vapor,  are  sub- 
ject to  the  same  physical  laws  of  compression  and  dilatation,  as 
the  permanent  gases.  Accordingly,  let  to  represent  the  weight 
of  a  cubic  inch  of  aqueous  vapor,  such  as  It  would  be,  if  it 
were  capable  of  being  reduced  to  the  temperature  of  melting 
ice,  and  under  a  barometric  pressure  of  29,92  inches ;  *  and  let 
u/  represent  the  weight  of  an  equal  volume  of  the  same  vapor, 
as  it  would  really  exist  at  another  temperature  t,  and  with  the 
elastic  force  F,  equal  or  inferior  to  the  maximum  density  be- 
longing to  this  temperature.  If  we  put  t'  =  t  —  32,  or  the 
number  of  degees  distant  from  freezing,  according  to  the  laws  of 
470.     the  dilatation  of  the  permanent  gases,  we  shall  have 

,  _   wl^ 

^   ~   29,92  (1  -j-  «'.  0,00208)' 

All  the  experiments  of  philosophers  upon  aqueous  vapor,  and 
those  of  Despretz  upon  the  vapor  of  several  other  liquids, 
show  that  this  formula  is  really  applicable  to  them  ;  so  that  we 
may  safely  employ  it  for  the  purpose  under  consideration.  Now 
calling  c'  the  quantity  of  heat  necessary  to  convert  15,5  grains 
Troy  t  of  water,  in  a  liquid  state,  at  32^,  into  vapor,  having  the 
temperature  t' ;  c'  w'  will  be  the  absolute  quantity  of  heat  neces- 
sary to  form  the  weight  w'  of  such  a  vapor ;  and  consequently 
282.  if  n  be  the  number  of  cubic  inches  contained  in  the  cylinder 
to  be  filled,  the  whole  quantity  of  heat  necessary  for  this  purpose, 
will  be 

'  c'  n  ID  F 

29,92  (1  +  t'  .   0,00208)" 

*  Or,  more  accurately,  15,444  grains,  or  one  gramme, 
t  Equal  to  0,76  of  a  nnctre. 


Steam-Engine.  423 


•<b 


Having  thus  brought  into  an  equation  all  the  different  ele- 
ments from  which  this  expenditure  of  heat  results,  we  are  able 
to  analyze  the  several  effects  produced  by  them.  In  the  6rst 
place,  according  to  the  experiments  of  Clement,  Southern,  and 
Despretz,  before  mentioned,  d  is  sensibly  constant  at  all  tempe- 
ratures at  which  we  have,  yet  been  able  to  make  observations, 
and  accordingly  the  expenditures  will  be  always  the  same.  As 
to  the  elastic  force  F,  we  know  that  it  augments  according  as 
the  temperature  is  raised.  But  it  will  be  seen  by  tbe  formula 
that  the  density  iv',  and  consequently  the  expenditure  of  heat, 
increases  in  proportion  to  F.  Consequently  if  we  take  away  the 
factor  (I  -|-  ^'  •  0,0020S),  arising  from  the  dilatation  produced  by 
the  increase  of  temperature,  the  expenditure  necessary  to  pro- 
duce the  force  F  will  be  exactly  proportional  to  this  force,  and 
we  shall  neither  gain  nor  lose  any  thing  by  giving  it  a  greater 
or  less  energy.  But  the  influence  of  the  factor  (l  -\-  t'  .  0,00208,) 
which  increases  as  the  temperature  is  raised,  diminishes  the 
expenditure  in  question,  in  proportion  as  the  temperature  be- 
comes higher  ;  for  if  we  operate,  for  instance,  at  212°,  this  factor 
becomes  1,375  ;  and  if  at  320°,  it  becomes  1,600  ;  so  that 
the  relative  expenditure  in  this  last  case,  compared  with  that  in 
the  first,  with  the  same  elastic  force,  is  nearly  in  the  ratio  of 
1375  to  1600,  or  nearly  of  6  to  7.  Such  then  is  the  saving  of  heat 
that  may  be  made  in  the  formation  of  steam  in  the  extremes  of 
temperature  at  which  we  have  as  yet  operated.  Accordingly  if  it 
were  true  that  high-pressure  engines  have,  over  those  of  the  ordi- 
nary kind,  an  advantage  as  great  as  has  been  alleged,  it  must  be 
sought  in  something  else  beside  the  saving  of  fuel.  But  in  order 
to  judge  correctly  of  these  engines  it  is  necessary  to  take  into 
consideration  another  element,  namely,  the  ulterior  developement 
of  elastic  power,  which  the  steam  thus  formed  is  capable  of  fur- 
nishing, when,  after  having  been  employed  with  its  primitive 
energy  in  the  first  cylinder,  it  is  made  to  pass  into  another  larger 
cylinder  where  it  dilates,  undergoing  a  reduction  of  temperature 
at  the  same  time,  in  such  a  manner  as  always  to  fill  the  space 
into  which  it  is  received,  until  at  length  it  is  condensed  into  water, 
when  it  no  longer  has  an  elastic  force,  either  equal  or  inferior  to 
the  pressure  of  the  atmosphere.  It  is  manifest  that  this  ulterior 
developement  of  force  must,  in  order  that  the  whole  effect  may 
be    appreciated,    be    added    to    the    mechanical    power  at    the 


424  Hydrodynamics, 

commencement ;  and  it  is  no  less  evident  that  it  offers  a  peculiar 
advantage  to  engines  in  which  the  steam,  before  being  condensed, 
is  employed  at  a  high  temperature.  Nevertheless,  the  numerous 
experiments  which  have  been  made  within  a  few  years  upon 
engines  of  this  kind,  some  of  which  have  been  accompanied 
with  careful  measurements,  have  not  seemed  to  confirm,  so  much 
as  might  have  been  expected,  the  favorable  view  we  have  given 
above  ;  and  if  they  have  taught  us  a  real  saving  of  fuel,  when 
considered  with  reference  to  the  force  actually  developed,  this 
saving  does  not  appear  to  exceed  the  narrow  limits  just  assigned 
to  the  effect  of  the  heat  of  dilatation.  So  small  an  advantage 
will  be  far  from  compensating  for  all  the  additional  precautions 
required  in  such  engines,  the  dangers  incurred,  and  the  numer- 
ous causes  of  waste  to  which  they  are  liable.  It  is  necessary,  in 
the  first  place,  to  make  the  boilers,  cylinder,  &ic.,  very  strong, 
that  they  may  resist  the  expansive  force  exerted  by  the  steam. 
It  is  likewise  necessary  to  give  greater  perfection  to  the  pistons, 
and  to  apply  more  frequently  some  lubricating  substance  to 
preserve  the  contact.  Repairs  are  in  consequence  often  required, 
and   the  value  of  the  engine  is  sensibly  diminished  on   this  account. 

Attempts  have  been  made  to  remove  this  great  inconvenience 
by  an  expedient  formerly  employed  to  perfect  the  first  inv^ention 
of  Savary.  It  is  this ;  the  piston,  instead  of  being  in  immediate 
contact  with  the  aqueous  vapor,  which  melts  and  dissolves  the 
grease  with  w'hich  it  is  impregnated,  receives  its  motion  through 
the  intervention  of  a  column  of  oil,  or  some  other  unctuous  sub- 
stance not  easily  evaporated,  on  which  the  steam  is  made  to  act 
by  pressure.  For  this  purpose  the  cylinder  in  which  the  piston 
plays  is  enclosed  in  a  larger  cylinder  with  which  it  communf- 
cates,  and  which  contains  the  oil.  The  oil  rising  and  falling 
continually  in  the  interior  cylinder  keeps  it  always  lubricated. 
But  although  this  ingenious  arrangement  may  be  sometimes 
adopted  with  advantage,  it  could  not  be  used  at  high  tempera- 
tures ;  for,  according  to  the  judicious  observation  of  AJr.  Watt, 
the  oil  would  be  decomposed  by  the  dissolving  power  of  the 
steam. 

It  has  likewise  been  proposed  to  construct  high-pressure 
engines  having  a  very  great  power  with  very  little  bulk,  by  em- 
ploying   small    boilers,    made    so    strong    as    to   resist    the    most 


Steam-Engine.  ^5 

powerful  pressure,  while  ihey  admit  of  being  heated,  as  it  were 
red  hot ;  whereby  steam  would  be  obtained  of  an  excessively 
high  temperature,  and  which,  developing  itself  by  its  expansive 
force  in  the  great  cylinder,  would  possess  even  after  its  dilata- 
tion an  elastic  force  sufficiently  energetic.  A  similar  arrange- 
ment was  likewise  attempted  formerly,  in  order  to  furnish  steam 
for  Savary's  engine ;  but  it  does  not  appear  to  have  been  found 
profitable  enough  to  be  continued  in  use;  and  after  the  calcula- 
tions we  have  made  on  the  expenditure  of  heat  employed  in  the 
formation  of  steam  at  every  temperature,  it  seems  improbable 
that  a  sufficient  saving  can  be  made  in  such  engines  to  compen- 
sate for  their  inconveniences.  Advantages  of  a  different  kind 
might  be  realized,  if  we  could  succeed  in  employing,  without  loss, 
some  liquid  different  from  water,  having  a  much  greater  elastic 
force  at  the  same  temperature. 

One  inevitable  consequence  of  employing  high  temperatures 
is,  that  the  loss  of  heat  by  radiation  is  much  greater,  and  this 
makes  an  important  item  in  calculating  the  results.  To  form 
an  idea  of  the  diminution  of  effect  arising  from  these  different 
circumstances,  we  are  to  remember  that  according  to  Lavoisier 
and  Laplace,  1  gramme  or  15,444  grains  Troy  of  charcoal,  devel- 
opes,  in  burning,  13038  degrees  of  heat  by  Fahrenheit's  scale,  or 
about  92°  on  Wedgewood's.  Now  15,444  grains  of  water  at  the 
temperature  of  212°,  by  being  converted  into  steam  absorb 
1020,6°  by  Fahrenheit;  then  15,444  grains  of  charcoal  would 
reduce  to  steam  200  grains  of  water,  on  the  supposition  that  no  heat 
is  lost,  and  that  the  water  is  already  brought  to  the  temperature 
of  212°.  But  after  a  great  number  of  experiments  made  upon  the 
most  perfect  engines  and  the  best  constructed  furnaces,  Mr.  Clem- 
ent found  that  1544,4  grains  of  charcoal  does  not  produce  more 
than  G  or  7  limes  as  much  steam,  and  1544,4  grains  of  the  best 
fossil  coal  never  produces  more  than  6  times  as  much  steam; 
whence  it  will  be  seen  that  nearly  half  the  heat  is  lost  by  radia- 
tion, and  the  conducting  power  of  the  boiler  and  the  surrounding 
bodies.  The  loss  is  without  doubt  still  more  considerable  in  en- 
gines of  a  high  pressure. 

526.   When  we  know  the  elastic  force  of  the  steam  introduced 
under  the  surface  of   the  piston,   it  is  easy  to  estimate  the  whole 
pressure  resulting  from  it ;  but  in  this  estimate  it  is  necessary  to 
Mech.  54 


426  Hydrodynamics. 

take  account  of  the  tension  of  the  steam  which  remains  on   the 
other  side  when  the  vacuum  is  not  perfect.     After  all,  this  esti- 
mate  fails  of  giving  the  primitive  energy  of  the  power   employed, 
and  much  less  the  part  which  remains  to  be  disposed  of  after  all  the 
friction  is  overcome  and  the  different  parts  of  the    machine   are  put 
in  motion.     This  useful  part  of  the  force  can  be  measured  a  poste- 
riori by  the    effect  which   the  whole    engine   produces.     Ordina- 
rily  we  compare   the   effective  power  of  an   engine  with  that  of  a 
certain   number  of  horses  of  a  medium  strength,  and   the  force   of 
the  engine  is  estimated   accordingly.     By  a   great  number  of  ex- 
periments of  this  kind    Watt  and   Bolton  supposed    that   a  horse 
of  ordinary  strength,  working  8  hours  a  day,  would  raise    3200  lb. 
avoirdupois    one    foot    per    hour.     Smeaton    made    the    estimate 
2300,  and    Clement  about     1300.       If,  therefore,   we  divide   the 
number  of  pounds  that  a   steam-engine   will   raise    one  foot   (or, 
which  is  the   same   thing,  the  product   of  the  number  of  pounds 
into  the  number  of  feet  elevation),   by  the  number  representing  a 
horse-power,  the   quotient   will   be   the   number  of  horses  to  which 
the  engine  is  equivalent.     There    are   engines  of  the   power  of  20, 
30,   &;c.,  horses.     The  most  powerful   engine  that    has  yet  been 
constructed  is  that  employed   in  the  mines  of  Cornwall.     It  has  the 
power  of  1010  horses,  and  serves  to  drain   by   pumps  a  mine   590 
feet   deep.     It  is  evident,  that  the  power  in  question  is  all  that 
needs  to  be   estimated  ;  for  we  can   apply  it  to   the   raising  of  wa- 
ter, the  turning  of  spindles,  or  to  any  other  purpose  that  requires 
such  a  force.     The  transmission  of  the  primitive  motion  can  be 
effected  by  mechanical  means  and  instruments  of  which  we  have 
already  spoken,  and  which  need  not  be  again  described. 


NOTES. 


I. 


On  the  Measure  of  Forces. 

The  name  of  living  forces  has  been  applied  to  the  forces  of 
bodies  in  motion,  and  that  of  dead  forces  to  those,  which,  like  a  sim- 
ple pressure,  suppose  no  actual  motion  in  the  operating  cause. 

There  was,  for  a  considerable  time,  a  difference  of  opinion  among 
mathematicians  in  regard  to  the  measure  of  living  forces,  or  the 
forces  of  bodies  in  motion.  Some  affirmed  that  these  forces  ought 
not  to  be  measured  by  the  product  of  the  mass  into  the  velocity, 
according  to  the  rule  that  has  been  given,  but  by  the  product  of  the 
mass  into  the  square  of  the  velocity.  As  this  difference  in  the 
estimate  of  forces  may  be  deemed  of  great  importance  in  mechanics, 
it  will  be  proper  to  make  a  few  remarks  upon  it. 

It  is  altogether  unimportant  whether  we  measure  the  force  of 
bodies  in  motion  by  the  product  of  the  mass  into  the  velocity  simply, 
or  by  the  product  of  the  mass  into  the  square  of  the  velocity,  pro- 
vided that  in  the  two  cases  we  assign  different  significations  to  the 
word  force.  When  we  assume  as  the  measure  of  force,  the  product 
of  the  mass  into  the  square  of  the  velocity,  we  mean  by  the  term 
force  the  number  of  obstacles  which  a  moving  body  is  capable  of 
overcoming  ;  it  is  certain  that  the  number  of  obstacles  which  mov- 
ing bodies  of  equal  masses  are  capable  of  overcoming  is  proportional 
to  the  squares  of  the  velocities.  For  example,  if  the  body  A  has  Fig.  255. 
precisely  the  velocity  necessary  to  close  the  spring  ACB,  an  equal 
body  M  will  require  only  double  this  velocity  to  close  four  springs, 
each  equal  to  ACB.  For,  in  the  first  instant,  for  example,  the  body 
M  advancing  with  a  velocity  double  that  of  A,  will  close  the  four 
springs,  considered  as  one,  twice  as  much  as  A  would  close  its  sin- 
gle spring ;  each  of  the  four  springs  then  will  be  closed  only  half 
as  much  as  ACB,  and  consequently  will  have  opposed  during  this 
instant  a  resistance  only  half  as  great  as  that  of  ^  C£  ;  all  the  four 


428  JVotes. 

springs,  therefore,  in  being  each  reduced  the  same  angular  quantity 
as  ACD  is  reduced,  will  oppose  only  double  the  resistance.  In  the 
same  manner  it  may  be  demonstrated  that,  in  the  succeeding  in- 
stants, the  resistance  opposed  by  the  four  springs  in  being  closed 
each  the  same  angular  quantity,  as  that  by  which  ACB  is  closed  in 
one  instant,  is  always  to  that  opposed  by  the  single  spring  ACB,  in 
the  same  instant,  in  the  ratio  of  the  velocities  belonging  to  the  two 
bodies  ;  therefore  a  velocity  double  that  required  to  close  one  spring 
is  sufficient  to  close  four  springs.  Thus  the  number  of  springs 
closed,  which  are  1  and  4,  are  as  the  squares  of  the  velocities  1  and 
2  necessary  to  close  them. 

We  see  then  that  the  number  of  obstacles  which  bodies  in  mo- 
tion are  capable  of  overcoming  increases  as  to  the  squares  of  the 
velocities.  But  by  the  term  force  ought  we  to  understand  the  niim- 
her  of  obstacles  1  Is  it  not  much  more  natural  to  consider  it  as  de- 
noting the  sum  of  the  resistances  opposed  by  these  obstacles  ?  For 
it  is  not  merely  the  number,  but  the  value  of  each  obstacle  which 
destroys  motion.  Now  in  this  case  each  instantaneous  resistance 
being  evidently  proportional  to  the  quantity  of  motion  destroyed  by  it, 
(on  which  point  all  are  agreed,)  the  sum  of  the  resistances  will  be 
proportional  to  the  quantity  of  motion  destroyed.  If  then  by  force, 
we  understand  the  sum,  and  not  merely  the  number  of  the  resistances 
which  a  moving  body  is  capable  of  overcoming,  the  force  is  propor- 
tional to  the  quantity  of  motion.  From  this  principle  it  has  likewise 
been  inferred  that  the  number  of  resistances  overcome  are  as  the 
squares  of  the  velocities.  The  question  then  is  in  reality  nothing 
more  nor  less  than  a  question  about  terms,  and  reduces  itself  to  find- 
ing the  meaning  of  the  word  force.  As  to  this  point  we  are  per- 
fectly at  liberty ;  provided  we  employ  that  which  we  take  for  the 
measure  of  force  agreeably  to  the  idea  which  we  attach  to  the 
term  force,  we  shall  always  arrive  at  the  same  results.  We  shall, 
therefore,  continue  to  take  for  the  measure  of  forces  the  product  of 
the  mass  into  the  velocity ;  and  consequently  by  the  force  of  a  body 
we  understand  the  sura  total  of  the  resistances  necessary  to  exhaust 
its  motion. 

II. 

On  the  Compressibility  of  Water. 

The  phenomenon  of  the  transmission  of  sound  through  water  and 
other  liquids  had  long  indicated  that  they  were  capable  of  being 
compressed.     Canton,   an  English  philosopher,  clearly  detected  this 


J^otes.  429 

property  by  observing  the  volume  occupied  respectively  by  oil, 
water,  and  mercury,  first  placed  in  a  vacuum,  and  afterwards  ex- 
posed to  the  pressure  of  the  atmosphere ;  but  the  results  which 
he  obtained,  though  exact  in  themselves,  were,  however,  liable 
to  be  affected  by  the  accidental  variations  of  form  and  tempera- 
ture to  which  the  apparatus  was  subject.  M.  Oersted  completely 
removed  these  difficulties  by  plunging  the  liquid  to  be  compressed, 
together  with  the  vessel  containing  it,  into  another  liquid  to  which 
the  pressure  was  applied,  and  through  which  it  was  made  to  pass  to 
the  interior  liquid  without  changing  the  form  of  the  vessel,  since  it 
acted  equally  within  and  without.  M.  Oersted  found,  likewise,  that  a 
pressure  equal  to  the  weight  of  the  atn)osphere  produces  in  pure 
water  a  diminution  of  volume  equal  to  0,000045  of  its  original  vol- 
ume. The  experiments  of  Canton  gave  0,000044.  M.  Oersted  found, 
by  varying  the  pressure  from  ^  of  the  weight  of  the  atmosphere  to  6 
atmospheres,  a  change  of  volume  sensibly  proportional  to  the  pres- 
sure. Later  experiments,  made  by  Mr.  Perkins,  seem  to  show  that 
this  proportionality  continues  when  the  pressure  amounts  to  2000 
atmospheres.  Before  the  water,  however,  is  entirely  freed  from  air, 
,  the  diminution  of  volume,  produced  by  the  pressure,  is  at  first  some- 
what greater  than  the  above  ratio  would  indicate. 


III. 


On  the   Condensation  of  Gases  into  Liquids. 

Mr.  Faraday  enclosed  in  glass  tubes,  bent  and  sealed,  different 
chemical  products  which  were  capable  of  developing  gases  by  their 
mutual  combination.  He  introduced  them  into  the  tubes  in  such  a 
manner  that  they  remained  separate  in  the  different  branches  of 
each  tube,  and  were  not  mixed  until  the  tube  had  been  sealed.  All 
the  gas  developed  in  each  tube  was  found  to  be  confined  to  a  fixed 
volume,  to  which  it  could  be  reduced  only  by  the  action  of  a  con- 
siderable pressure  ;  this  pressure  causes  it  to  liquify  in  the  follow- 
ing cases.  1.  Sulphurous  acid  produced  by  the  action  of  sulphuric 
acid  on  mercury.  2.  Sulphuretted  hydrogen  produced  by  the  action 
of  hydrochloric  acid  on  the  sulphurct  of  iron  in  fragments,  3.  Car- 
bonic acid  produced  by  the  action  of  sulphuric  acid  on  carbonate  of 
ammonia.  4.  Oxide  of  chlorine,  produced  by  chlorate  of  potash 
and  sulphuric  acid.     5.  Ammonia  disengaged  from  the  combination 


430  JVotes. 

of  this  substance  with  chloride  of  silver,  &,c.  In  each  experiment 
the  branch  of  the  tube  containing  the  mixture  was  warmed,  while  the 
other  was  cooled  with  moistened  paper. 


IV. 


On  the  Construction  of  Valves. 

A  VALVE  is  a  kind  of  lid  or  cover  to  a  tube  or  vessel,  so  contrived 
as  to  open  one  way  by  the  impulse  of  any  fluid  against  it,  and  to 
close,  when  the  motion  of  the  fluid  is  in  the  opposite  direction,  like 
the  clapper  of  a  pair  of  bellows.  In  the  air-pump  this  purpose  is 
effected  by  means  of  a  strip  of  leather,  bladder,  or  oiled  silk, 
stretched  over  a  small  perforation  in  the  piston,  and  ordinarily  in  a 
507.    plate  at  the  bottom  of  the  barrel. 

In  common  water-pumps  the  valve,  or  sucker,  as  it  is  often  called, 
is  a  thick  piece  of  leather  pressed  down  by  a  small  wooden  weight, 
and  turning  on  the  flexible  leather  as  a  hinge.  It  is  represented  at 
E,  figure  245,  &c.  In  the  best  metallic  pumps  for  raising  water  the 
valve  consists  of  a  metallic  cone  with  a  stem  and  knob,  as  represented 
at  L  in  figure  247.  The  conical  part  is  ground  so  as  to  fit  accurately 
the  rim  of  the  opening  in  which  it  plays,  and,  unlike  other  valves,  it 
is  rendered  tighter  by  use,  and  is  less  likely  to  be  obstructed  by  for- 
eign substances  contained  in  the  water. 

Mr.  Perkins  invented  a  pump  having  a  square  bore,  in  which  the 
valve  consists  of  two  triangular  pieces  of  leather  loaded  with  weights, 
and  turning  on  a  metallic  hinge  placed  diagonally  across  the  bore. 
Mr.  Evans  adapted  a  similar  kind  of  valve  to  the  common  pump  of  a 
circular  bore,  the  form  of  the  valve  being  a  semi-ellipse.  The  chief 
advantage  of  this  construction  is,  that  there  is  very  little  obstruction 
to  the  motion  of  the  water,  and  consequently  less  loss  of  power,  than 
in  the  common  pump,  where  the  space,  left  for  the  passage  of  the 
water,  bears  a  less  proportion  to  the  whole  bore. 


Notes.  431 

V. 

On  the  History  and  Construction  of  the  Barometer. 

The  barometer  takes  its  origin  from  the  experiment  of  Torri- 
celli,  who  in  consequence  of  the  suggestion  of  Galileo  with  regard 
to  the  ascent  of  water  in  pumps,  proceeded  in  1643  to  make  experi- 
ments with  a  tube  filled  with  mercury,  conjecturing  that  as  this 
fluid  was  about  thirteen  times  heavier  than  water,  it  would  stand 
at  only  one  thirteenth  of  the  height  to  which  water  rises  in  pumps, 
or  at  about  thirty  inches.  He,  therefore,  filled  a  glass  tube  about 
three  feet  long  with  mercury,  and  upon  immersing  the  open  end  in 
a  vessel  of  the  same  fluid,  he  found  that  the  mercury  descended  in 
the  tube,  and  stood  at  about  twenty-nine  and  a  half  R,oman  inches, 
and  this  vertical  elevation  was  preserved,  whether  the  tube  was 
perpendicular  or  inclined  to  the  horizon,  according  to  the  known 
laws  of  hydrostatical  pressure.  This  celebrated  experiment  was 
repeated  and  diversified  in  several  ways  with  tubes  filled  with 
other  fluids,  and  the  result  was  the  same  in  all,  allowance  being 
made  for  difference  of  specific  gravity,  and  thus  the  weight  and 
pressure  of  the  air  were  fully  established.  Such,  however,  was  the 
force  of  prejudice  that  many  refused  to  yield  their  assent  till,  at  the 
suggestion  of  Pascal,  the  experiment  was  performed  at  different 
heights  in  the  air  with  such  results  as  left  no  longer  any  doubt  upon 
the  subject. 

Great  care  is  necessary  in  the  construction  of  the  barometer. 
The  tube  after  being  cleansed  as  perfectly  as  possible,  is  to  be  grad- 
ually heated,  and  to  be  kept  at  a  pretty  high  temperature  for  a  con- 
siderable time,  for  the  purpose  of  expelling  all  moisture  that  may 
be  found  adhering  to  it.  The  mercury  is  then  to  be  introduced ;  a 
small  quantity  only  is  first  poured  in  by  means  of  a  fine  funnel  and 
thoroughly  boiled  in  order  to  free  it  from  air  ;  then  another  portion 
is  added,  and  so  on  till  the  tube  is  filled.  It  is  afterward  to  be  care- 
fully inverted,  and  the  open  end  immersed  in  a  cistern  of  boiled 
mercury. 

The  tube  and  cistern  is  enclosed  in  a  metallic  or  wooden  frame- 
work, containing  the  graduations  and  some  necessary  appendages.  As 
the  mercury  rises  and  falls  in  the  tube  by  the  fluctuations  of  the 
.  atmosphere,  its  surface  varies  also  in  the  cistern.  But  the  gradua- 
tion is  intended  to  mark  the  exact  length  of  the  column,  reckoned 
from  this  variable  surface.  If  a  horizontal  section  of  the  tube  and 
cistern  have  a  constant  ratio  to  each  other   throughout   the  extent 


432  Notes. 

embraced  by  these  changes,  a  correction  could  be  readily  applied. 
Suppose,  for  instance,  that  a  section  of  the  cistern  is  one  hundred 
times  that  of  the  tube,  or  that  their  diameters  are  as  1  to  10,  and 
that  the  surface  of  the  mercury  in  the  cistern  coincides  with  the 
point  from  which  the  graduations  commence  when  the  mercury  in 
the  tube  stands  at  thirty  inches.  The  correction  would  be  one  hun- 
dreth  part  of  the  difference  from  30  inches,  and  additive  or  sub- 
tractive,  according  as  this  difference  was  below  or  above  30. 

It  is  usual,  however,  in  the  best  barometers  to  bring  the  surface 
of  the  mercury  in   the  cistern  to    the    point    from  which  the   gradua- 

Fig.  232.  tions  commence  by  means  of  a  screw  V,  acting  on  a  flexible  piece 
of  leather  which  forms  the  bottom  of  the  cistern.  We  are  able  to 
tell  when  the  desired  coincidence  is  effected  by  means  of  a  mark  on 
a  piece  of  ivory  floating  on  the  mercury  and  sliding  over  a  fixed 
object  having  a  corresponding  mark.  Sometimes  the  cistern  is  of 
glass,  and  the  point  of  commencement  of  the  graduations  is  marked 
upon  it,  or    (which  is  much  better)  is   indicated    by  the  contact  of  a 

Fig. 231.  sharp  ivory  pin  P,  inserted  in  the  cap,  and  descending  into  the  in- 
terior of  the  cistern. 

There  are  various  kinds  of  portable  barometers  constructed  for 
the  purpose  of  measuring  the  heights  of  mountains.  The  latest  and 
most  convenient  is  represented  in  figure  232.  It  is  of  the  syphon 
form,  and  was  invented  by  Gay-Lussac.  The  barometer  being  filled, 
the  extremity  of  the  shorter  branch  1"  is  hermetically  sealed.  In 
this  state  the  barometer  is  inaccessible  to  the  external  air,  and  con- 
sequently is  incapable  of  indicating  the  changes  of  pressure  in  the 
atmosphere  ;  but  to  open  the  communication,  we  draw  out,  by  means 
of  a  blow-pipe,  a  small  portion  of  the  glass  near  the  middle  of  the 
shorter  brancii,  on  the  inside,  and  form  a  fine  capillary  tube,  which  is 
sufficient  to  admit  the  air  but  does  not  allow  the  mercury  to  escape, 
on  account  of  the  force  with  which  it  repels  it  in  virtue  of  its  capil- 
lary action.  The  difference  of  level  between  the  two  extremities 
S,  N,  of  the  column  being  observed,  it  is  reversed,  and  a  part  of  the 
mercury  enters  the  longer  branch  CJC,  and  fills  it,  the  rest  falls  into 
the  shorter  branch  CY,  but  cannot  escape  for  the  reason  above  men- 
tioned. It  may  then  be  carried  in  this  position,  being  always  open 
to  the  air,  but  not  to  the  mercury. 

This  barometer  may  be  enclosed  in  a  cane  and  transported  with 
great  ease  and  safety.  A  small  thermometer  is  appended,  as  in 
other  cases,  for  the  purpose  of  measuring  the  temperature  of  the 
mercury.  By  contracting  the  tube  near  the  two  ends  we  prevent 
all  danger  of  its  breaking  by  any  sudden  motion  in  the  column  of 
mercury. 


JVotes.  433 

By  observing  regularly  the  height  of  the  barometer  for  a  con- 
siderable period  in  the  same  place,  we  find  that  it  does  not  remain 
constantly  the  same.  For  some  time  after  the  instrument  was  in- 
vented, it  was  supposed  that  the  mercury  stood  higher  just  before 
rain,  and  lower  during  fair  weather.  Reasons  were  assigned  for  this 
supposed  fact.  It  was  said,  that  when  it  is  about  to  rain,  the  air  is 
charged  with  water,  and  consequently  that  tHe  weight  of  the  atmo- 
sphere is  more  considerable  ;  and  that,  on  the  contrary,  this  weight 
must  be  less  in  fair  weather,  because  the  air  is  then  relieved  from 
a  certain  part  of  the  moisture  contained  in  it.  Unfortunately  for 
this  hypothesis  it  has  been  found,  more  recently,  that  the  quantity  of 
water  which  the  air  is  capable  of  containing,  increases  with  the 
temperature,  so  that  in  summer  it  contains,  for  the  most  part,  more 
water  than  in  winter,  although  there  is  less  fair  weather  in  winter 
than  in  summer.  It  appears  also  that  the  vapor  of  water  is  lighter 
than  the  same  volume  of  air,  when  the  same  elastic  force  is  exert- 
ed ;  that  is,  if  we  substitute  for  a  cubic  foot  of  air,  taken  at  a  certain 
height  in  the  atmosphere,  a  cubic  foot  of  aqueous  vapor,  of  the 
same  temperature  and  elasticity,  the  vapor  will  weigh  less  than 
the  air,  and  will  consequently  exert  less  pressure  upon  the  barome- 
ter. We  hence  draw  a  conclusion  the  reverse  of  that  which  the 
first  observers  of  the  barometer  undertook  to  maintain,  namely,  that 
the  rise  of  the  barometer  indicates  fair  weather,  and  its  fall,  rain. 
This  is  in  fact  agreeable  to  observation  in  ordinary  cases.  But  it 
must  be  confessed  that  the  reason  now  given  is  but  little  better  than 
that  we  have  been  combating. 

The  variations  of  the  barometer  are  different  in  different  places. 
They  are  almost  nothing  upon  the  tops  of  high  mountains,  and  be- 
tween the  tropics;  even  in  the  temperate  zones  they  are  never 
very  great  in  calm  weather.  But  the  barometer  almost  always  de- 
scends rapidly  before  a  violent  storm,  great  changes  taking  place  in 
a  few  hours.  On  this  account  the  instrument  is  particularly  useful 
at  sea. 

By  comparing  observations  made  at  different  and  remote  places, 
we  discover  a  remarkable  correspondence,  which  shows  a  simulta- 
neousness  in  the  motions  of  the  atmospheric  strata  that  would  hardly 
have  been  expected.  Still  this  correspondence  is  far  from  being 
perfect,  especially  as  to  the  quantity  of  the  change. 

By  examining   a  long  series   of  observations  made  in  the  same 

place,  we  shall  perceive,  amid  all  the  accidental  irregularities,   that 

there  is  a  general  tendency,  occurring  periodically,  to  rise   and  fall 

at  certain  hours  of  the  day.     By  a  long  series  of  observations,  direct- 

Mech.  65 


434  JVoies. 

ed  to  this  point,  M.  Raymond  discovered,  that  in  France  the  barome- 
ter attains  its  maximum  elevation  about  9  o'clock  A.  M.,  after  which 
it  descends  till  about  4  P.  ]M.,.when  it  is  at  its  minimum  ;  from  this 
time  it  rises  till  near  11  P.  M.,  when  it  reaches  its  maximum  again, 
after  which  it  commences  a  downward  motion  till  4  A.  M. ;  and 
thence  it  begins  to  return  to  the  state  first  mentioned.  This  march 
is  often  deranged  in  European  climates,  where  the  atmosphere  is  so 
variable ;  but  under  the  tropics  where  the  causes  which  act  upon 
the  atmosphere  are  more  constant,  the  periodical  changes  are  regu- 
lar, and  to  such  a  degree  that,  according  to  Humboldt,  one  may 
almost  predict  the  hour  of  the  change  at  any  time  from  a  single  ob- 
servation ;  and,  what  is  very  remarkable,  the^e  changes,  according 
to  the  same  distinguished  philosopher,  are  not  affected  by  any  at- 
mospherical circumstance ;  neither  the  wind,  nor  rain,  nor  fair 
weather,  nor  tempests,  disturb  the  perfect  regularity  of  these  oscilla- 
tions. They  are  found  to  be  the  same  in  all  weathers  and  at  all  sea- 
sons. For  further  particulars  relative  to  the  construction  of  the  ba- 
rometer and  the  theory  of  its  fluctuations,  the  student  is  referred  to 
Daniell's  Meteorological  Essays. 


THE    END. 


.Morli.iliws 


(     .lH!.! 


27 

?    c 

B   «t" 

i' 

7/ 

i 

B  ; 

---    G 

,„ 

!k 

|a 

■  --^( 

9.0 


© ©-S— © 


J7 

A  B_ 


J^ 


I 

/ 

(' 

'>'.)  :^ 


(Minl.pi.l...-    Xal.J'Inl 


.M<<  li.uuj  s  i'/  .U  . 


'26 


r     r      B' G"        J 


1) 

a7 

m 

X 

"/i ' 

■^A 

T 

c. 

S 

/ 

A' 

Cl/ 

/ 

2/) 


34^ 

C     X    i) 


A  li  B 


T^^' 


/<J 

-^ 

?!/- 

P         \ 

''/fv.-_ 



7i 

VJ 

r^ 

A 

/ 

Op- 

r." 

\ 

IVl/ 

p 

K 

\ 

«  .- 

*.' 

.., 

'"/ 

/' 

JJ         I.  (V  ('  A" 


.52 


yy 


\ 

K 

K 

T 
\ 

^. 

.///, 


66 


m 


Ciiitiliiirl...-     Snt    I'ljil. 


M>-.l,.,ni,.s    /■////. 


t  ;»iitl»ri.|.>>-    X;il  .  I'liil. 


Mri'luui<\s  /"/./r 


ii..Im    i.U..'     X.ll.   I'llil 


Ml",  hiini.s   /•/     I'. 


Mecliiuiics    /*/  •  T7.      y 


l'(S' 


(  .;'      ;B 


r// 


V.     1 


iliiUlo..  Xat    Phil 


Jrediauirs     I'J .   17. 


It.  I> 


.s  tivn. 


^' 


—     c 


C'aiijJ.ii(l..i-   Xiil    Hiil 


jicMhauirs  ri.m. 


h  ""l7L 


hsi 

F 

IS'J 

K 

.      41 

1 

'Q 

III 

«-4|^ 

)' 

A  - 

At. 

u 

1 

Mf  ell  allies  Fl  IX. 


i>1hl.rlilue     Kat  ■  rilll  . 


Merlianics  Tl  H . 


'1 


•)  ')  '1 

It 

r 

K 

In" 

'.'/? 

\ 

( 

^^.-^i 

\ 

r 

B~ 

1, 
1'. 

.Mechanics  /'/.  .1  . 


•ill...-   v.ii    riliU 


V<?*fui.'  ^?M 


0CSB  LIBRART 

X-  SS  /^'/ 


JC  S01JT"ERN  REGIONAL  LBRARY  FACILITY 


A     000  606  495     0 


